# User:Jon Awbrey/Differential Logic Archive 2003–2004

MyWikiBiz, Author Your Legacy — Thursday May 19, 2022

Jump to navigationJump to search## Differential Logic 2003–2004

DLOG A. Differential Logic -- Series A DLOG B. Differential Logic -- Series B DLOG C. Differential Logic -- Series C DLOG D. Differential Logic -- Series D

### Differential Logic • Series A

#### DLOG 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(x, y) having the abstract type f : B x 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.

#### DLOG A • 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 DeMorgan'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-DeMorgan correspondence and 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(x, y) = 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 x B -> B or f : B^2 -> B. The concrete type takes into account the qualitative dimensions or the "units" of the case, which can be explained as follows. 1. Let X be the set of values {(x), x} = {not x, x}. 2. 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 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: 1. The "difference" operator [capital Delta], written here as D. 2. The "enlargement" operator [capital Epsilon], 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 x Y to its "differential extension", EU = U x dU = X x Y x dX x 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 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 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 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 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 & y => Df|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.

#### DLOG A • Note 3

Last time we computed what will variously be called the "difference map", the "difference proposition", or the "local proposition" Df_p for the proposition f(x, y) = xy at the point p where x = 1 and y = 1. In the universe U = X 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 Df_p = Df|p = Df|<1, 1> = Df|xy, so long as we know the frame of reference in force. Sticking with the example f(x, y) = 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 x Y x dX x 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(x, y).

#### DLOG A • Note 4

We have been studying the action of the difference operator D, also known as the "localization operator", on the proposition f : X 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 x Y x dX x 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 x Y x dX x 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 x Y and whose arrows are labeled with the elements of dU = dX x 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.

#### DLOG A • 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(x, y) = xy. Introduce a suitably generic definition of the extended universe of discourse: Let U = X_1 x ... x X_k and EU = U x dU = X_1 x ... x X_k x dX_1 x ... x dX_k. For a proposition f : X_1 x ... x X_k -> B, the (first order) "enlargement" of f is the proposition Ef : EU -> B that is defined by: Ef(x_1, ..., x_k, dx_1, ..., dx_k) = f(x_1 + dx_1, ..., x_k + dx_k). It should be noted that the so-called "differential variables" dx_j are really just the same kind of boolean variables as the other x_j. 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 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 Ef_xy + x(y) Ef_x(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.

#### DLOG A • Note 6

To broaden our experience with simple examples, let us now contemplate the sixteen functions of concrete type X x Y -> B and abstract type B x 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 o---------o---------o---------o----------o------------------o----------o | L_1 | L_2 | L_3 | L_4 | L_5 | L_6 | | | | | | | | | Decimal | Binary | Vector | Cactus | English | Ordinary | o---------o---------o---------o----------o------------------o----------o | | x : 1 1 0 0 | | | | | | y : 1 0 1 0 | | | | o---------o---------o---------o----------o------------------o----------o | | | | | | | | f_0 | f_0000 | 0 0 0 0 | () | false | 0 | | | | | | | | | f_1 | f_0001 | 0 0 0 1 | (x)(y) | neither x nor y | ~x & ~y | | | | | | | | | f_2 | f_0010 | 0 0 1 0 | (x) y | y and not x | ~x & y | | | | | | | | | f_3 | f_0011 | 0 0 1 1 | (x) | not x | ~x | | | | | | | | | f_4 | f_0100 | 0 1 0 0 | x (y) | x and not y | x & ~y | | | | | | | | | f_5 | f_0101 | 0 1 0 1 | (y) | not y | ~y | | | | | | | | | f_6 | f_0110 | 0 1 1 0 | (x, y) | x not equal to y | x + y | | | | | | | | | f_7 | f_0111 | 0 1 1 1 | (x y) | not both x and y | ~x v ~y | | | | | | | | | f_8 | f_1000 | 1 0 0 0 | x y | x and y | x & y | | | | | | | | | f_9 | f_1001 | 1 0 0 1 | ((x, y)) | x equal to y | x = y | | | | | | | | | f_10 | f_1010 | 1 0 1 0 | y | y | y | | | | | | | | | f_11 | f_1011 | 1 0 1 1 | (x (y)) | not x without y | x => y | | | | | | | | | f_12 | f_1100 | 1 1 0 0 | x | x | x | | | | | | | | | f_13 | f_1101 | 1 1 0 1 | ((x) y) | not y without x | x <= y | | | | | | | | | f_14 | f_1110 | 1 1 1 0 | ((x)(y)) | x or y | x v y | | | | | | | | | f_15 | f_1111 | 1 1 1 1 | (()) | true | 1 | | | | | | | | o---------o---------o---------o----------o------------------o----------o 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.

#### DLOG A • 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 : (U -> B) -> (EU -> B), 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: E_ij : (U -> B) -> (U -> B), E_ij : f -> E_ij f, E_ij f = 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 E_ij 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 E_ij, so the gang of four operators {E_ij : 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 T_00, T_01, T_10, T_11, 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 Z_4 (German "Zyklus"), which this is not. The other is Klein's four-group V_4 (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, T_00 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!

#### DLOG A • 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 X_1 x ... x X_k -> Y_1 x ... x Y_n and abstract types B^k -> B^n. 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 x Y x Z -> B and abstract type f : B^3 -> 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 u_1 ... u_k, where u_j = x_j or u_j = (x_j), for j = 1 to k. The proposition (u_1, ..., u_k) indicates the disjunctive region consisting of the cells that are "just next door" to the cell u_1 ... u_k.

#### DLOG A • 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" V_4. 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 E_ij operators and the T_ij 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.

#### DLOG A • 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, page 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.

#### DLOG A • 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, 'Collected Papers', 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.

#### DLOG A • 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.

#### DLOG A • 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).

#### DLOG A • 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.

#### DLOG A • 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.

#### DLOG A • 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.

#### DLOG A • 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).

#### DLOG A • 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.

#### DLOG A • 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, let's return to the recipe that we worked out before: 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.

#### DLOG A • 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

#### DLOG A • 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 have made us accustomed had he been given 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.

### Differential Logic • Series B

#### DLOG B • Note 1

| The most fundamental concept in cybernetics is that of "difference", | either that two things are recognisably different or that one thing | has changed with time. | | William Ross Ashby, |'An Introduction to Cybernetics', | Chapman & Hall, London, UK, 1956, | Methuen & Company, London, UK, 1964. Linear Topics. The Differential Theory of Qualitative Equations This chapter is titled "Linear Topics" because that is the heading under which the derivatives and the differentials of any functions usually come up in mathematics, namely, in relation to the problem of computing "locally linear approximations" to the more arbitrary, unrestricted brands of functions that one finds in a given setting. To denote lists of propositions and to detail their components, we use notations like: !a! = <a, b, c>, !p! = <p, q, r>, !x! = <x, y, z>, or, in more complicated situations: x = <x_1, x_2, x_3>, y = <y_1, y_2, y_3>, z = <z_1, z_2, z_3>. In a universe where some region is ruled by a proposition, it is natural to ask whether we can change the value of that proposition by changing the features of our current state. Given a venn diagram with a shaded region and starting from any cell in that universe, what sequences of feature changes, what traverses of cell walls, will take us from shaded to unshaded areas, or the reverse? In order to discuss questions of this type, it is useful to define several "operators" on functions. An operator is nothing more than a function between sets that happen to have functions as members. A typical operator F takes us from thinking about a given function f to thinking about another function g. To express the fact that g can be obtained by applying the operator F to f, we write g = Ff. The first operator, E, associates with a function f : X -> Y another function Ef, where Ef : X x X -> Y is defined by the following equation: Ef(x, y) = f(x + y). E is called a "shift operator" because it takes us from contemplating the value of f at a place x to considering the value of f at a shift of y away. Thus, E tells us the absolute effect on f that is obtained by changing its argument from x by an amount that is equal to y. Historical Note. The protean "shift operator" E was originally called the "enlargement operator", hence the initial "E" of the usual notation. The next operator, D, associates with a function f : X -> Y another function Df, where Df : X x X -> Y is defined by the following equation: Df(x, y) = Ef(x, y) - f(x), or, equivalently, Df(x, y) = f(x + y) - f(x). D is called a "difference operator" because it tells us about the relative change in the value of f along the shift from x to x + y. In practice, one of the variables, x or y, is often considered to be "less variable" than the other one, being fixed in the context of a concrete discussion. Thus, we might find any one of the following idioms: 1. Df : X x X -> Y, Df(c, x) = f(c + x) - f(c). Here, c is held constant and Df(c, x) is regarded mainly as a function of the second variable x, giving the relative change in f at various distances x from the center c. 2. Df : X x X -> Y, Df(x, h) = f(x + h) - f(x). Here, h is either a constant (usually 1), in discrete contexts, or a variably "small" amount (near to 0) over which a limit is being taken, as in continuous contexts. Df(x, h) is regarded mainly as a function of the first variable x, in effect, giving the differences in the value of f between x and a neighbor that is a distance of h away, all the while that x itself ranges over its various possible locations. 3. Df : X x X -> Y, Df(x, dx) = f(x + dx) - f(x). This is yet another variant of the previous form, with dx denoting small changes contemplated in x. That's the basic idea. The next order of business is to develop the logical side of the analogy a bit more fully, and to take up the elaboration of some moderately simple applications of these ideas to a selection of relatively concrete examples.

#### DLOG B • Note 2

Example 1. A Polymorphous Concept I start with an example that is simple enough that it will allow us to compare the representations of propositions by venn diagrams, truth tables, and my own favorite version of the syntax for propositional calculus all in a relatively short space. To enliven the exercise, I borrow an example from a book with several independent dimensions of interest, 'Topobiology' by Gerald Edelman. One finds discussed there the notion of a "polymorphous set". Such a set is defined in a universe of discourse whose elements can be described in terms of a fixed number k of logical features. A "polymorphous set" is one that can be defined in terms of sets whose elements have a fixed number j of the k features. As a rule in the following discussion, I will use upper case letters as names for concepts and sets, lower case letters as names for features and functions. The example that Edelman gives (1988, Fig. 10.5, p. 194) involves sets of stimulus patterns that can be described in terms of the three features "round" 'u', "doubly outlined" 'v', and "centrally dark" 'w'. We may regard these simple features as logical propositions u, v, w : X -> B. The target concept Q is one whose extension is a polymorphous set Q, the subset Q of the universe X where the complex feature q : X -> B holds true. The Q in question is defined by the requirement: "Having at least 2 of the 3 features in the set {u, v, w}". Taking the symbols u = "round", v = "doubly outlined", w = "centrally dark", and using the corresponding capitals to label the circles of a venn diagram, we get a picture of the target set Q as the shaded region in Figure 1. Using these symbols as "sentence letters" in a truth table, let the truth function q mean the very same thing as the expression "{u and v} or {u and w} or {v and w}". o-----------------------------------------------------------o | X | | | | o-------------o | | / \ | | / \ | | / \ | | / \ | | / \ | | o o | | | U | | | | | | | | | | | | | | | | | | | o--o----------o o----------o--o | | / \%%%%%%%%%%\ /%%%%%%%%%%/ \ | | / \%%%%%%%%%%o%%%%%%%%%%/ \ | | / \%%%%%%%%/%\%%%%%%%%/ \ | | / \%%%%%%/%%%\%%%%%%/ \ | | / \%%%%/%%%%%\%%%%/ \ | | o o--o-------o--o o | | | |%%%%%%%| | | | | |%%%%%%%| | | | | |%%%%%%%| | | | | V |%%%%%%%| W | | | | |%%%%%%%| | | | o o%%%%%%%o o | | \ \%%%%%/ / | | \ \%%%/ / | | \ \%/ / | | \ o / | | \ / \ / | | o-------------o o-------------o | | | | | o-----------------------------------------------------------o Figure 1. Polymorphous Set Q In other words, the proposition q is a truth-function of the 3 logical variables u, v, w, and it may be evaluated according to the "truth table" scheme that is shown in Table 2. In this representation the polymorphous set Q appears in the guise of what some people call the "pre-image" or the "fiber of truth" under the function q. More precisely, the 3-tuples for which q evaluates to true are in an obvious correspondence with the shaded cells of the venn diagram. No matter how we get down to the level of actual information, it's all pretty much the same stuff. Table 2. Polymorphous Function q o---------------o-----------o-----------o-----------o-------o | u v w | u & v | u & w | v & w | q | o---------------o-----------o-----------o-----------o-------o | | | | | | | 0 0 0 | 0 | 0 | 0 | 0 | | | | | | | | 0 0 1 | 0 | 0 | 0 | 0 | | | | | | | | 0 1 0 | 0 | 0 | 0 | 0 | | | | | | | | 0 1 1 | 0 | 0 | 1 | 1 | | | | | | | | 1 0 0 | 0 | 0 | 0 | 0 | | | | | | | | 1 0 1 | 0 | 1 | 0 | 1 | | | | | | | | 1 1 0 | 1 | 0 | 0 | 1 | | | | | | | | 1 1 1 | 1 | 1 | 1 | 1 | | | | | | | o---------------o-----------o-----------o-----------o-------o With the pictures of the venn diagram and the truth table before us, we have come to the verge of seeing how the word "model" is used in logic, namely, to distinguish whatever things satisfy a description. In the venn diagram presentation, to be a model of some conceptual description !F! is to be a point x in the corresponding region F of the universe of discourse X. In the truth table representation, to be a model of a logical proposition f is to be a data-vector !x! (a row of the table) on which a function f evaluates to true. This manner of speaking makes sense to those who consider the ultimate meaning of a sentence to be not the logical proposition that it denotes but its truth value instead. From the point of view, one says that any data-vector of this type (k-tuples of truth values) may be regarded as an "interpretation" of the proposition with k variables. An interpretation that yields a value of true is then called a "model". For the most threadbare kind of logical system that we find residing in propositional calculus, this notion of model is almost too simple to deserve the name, yet it can be of service to fashion some form of continuity between the simple and the complex. | Edelman, Gerald M., |'Topobiology: An Introduction to Molecular Embryology', | Basic Books, New York, NY, 1988.

#### DLOG B • Note 3

| The present is big with the future. | | ~~ Leibniz Here I now delve into subject matters that are more specifically logical in the character of their interpretation. Imagine that we are sitting in one of the cells of a venn diagram, contemplating the walls. There are k of them, one for each positive feature x_1, ..., x_k in our universe of discourse. Our particular cell is described by a concatenation of k signed assertions, positive or negative, regarding each of these features, and this description of our position amounts to what is called an "interpretation" of whatever proposition may rule the space, or reign on the universe of discourse. But are we locked into this interpretation? With respect to each edge x of the cell we consider a test proposition dx that determines our decision whether or not we will make a difference in how we stand regarding to x. If dx is true then it marks our decision, intention, or plan to cross over the edge x at some point within the purview of the contemplated plan. To reckon the effect of several such decisions on our current interpretation, or the value of the reigning proposition, we transform that position or that proposition by making the following array of substitutions everywhere in its expression: 1. Substitute "( x_1 , dx_1 )" for "x_1" 2. Substitute "( x_2 , dx_2 )" for "x_2" 3. Substitute "( x_3 , dx_3 )" for "x_3" ... k. Substitute "( x_k , dx_k )" for "x_k" For concreteness, consider the polymorphous set Q of Example 1 and focus on the central cell, specifically, the cell described by the conjunction of logical features in the expression "u v w". o-----------------------------------------------------------o | X | | | | o-------------o | | / \ | | / \ | | / \ | | / \ | | / \ | | o o | | | U | | | | | | | | | | | | | | | | | | | o--o----------o o----------o--o | | / \%%%%%%%%%%\ /%%%%%%%%%%/ \ | | / \%%%%%%%%%%o%%%%%%%%%%/ \ | | / \%%%%%%%%/%\%%%%%%%%/ \ | | / \%%%%%%/%%%\%%%%%%/ \ | | / \%%%%/%%%%%\%%%%/ \ | | o o--o-------o--o o | | | |%%%%%%%| | | | | |%%%%%%%| | | | | |%%%%%%%| | | | | V |%%%%%%%| W | | | | |%%%%%%%| | | | o o%%%%%%%o o | | \ \%%%%%/ / | | \ \%%%/ / | | \ \%/ / | | \ o / | | \ / \ / | | o-------------o o-------------o | | | | | o-----------------------------------------------------------o Figure 1. Polymorphous Set Q The proposition or the truth-function q that describes Q is: (( u v )( u w )( v w )) Conjoining the query that specifies the center cell gives: (( u v )( u w )( v w )) u v w And we know the value of the interpretation by whether this last expression issues in a model. Applying the enlargement operator E to the initial proposition q yields: (( ( u , du )( v , dv ) )( ( u , du )( w , dw ) )( ( v , dv )( w , dw ) )) Conjoining a query on the center cell yields: (( ( u , du )( v , dv ) )( ( u , du )( w , dw ) )( ( v , dv )( w , dw ) )) u v w The models of this last expression tell us which combinations of feature changes among the set {du, dv, dw} will take us from our present interpretation, the center cell expressed by "u v w", to a true value under the target proposition (( u v )( u w )( v w )). The result of applying the difference operator D to the initial proposition q, conjoined with a query on the center cell, yields: ( (( ( u , du )( v , dv ) )( ( u , du )( w , dw ) )( ( v , dv )( w , dw ) )) , (( u v )( u w )( v w )) ) u v w The models of this last proposition are: 1. u v w du dv dw 2. u v w du dv (dw) 3. u v w du (dv) dw 4. u v w (du) dv dw This tells us that changing any two or more of the features u, v, w will take us from the center cell to a cell outside the shaded region for the set Q.

#### DLOG B • Note 4

| It is one of the rules of my system of general harmony, | 'that the present is big with the future', and that he | who sees all sees in that which is that which shall be. | | Leibniz, 'Theodicy' | | Gottfried Wilhelm, Freiherr von Leibniz, |'Theodicy: Essays on the Goodness of God, | The Freedom of Man, & The Origin of Evil', | Edited with an Introduction by Austin Farrer, | Translated by E.M. Huggard from C.J. Gerhardt's | Edition of the 'Collected Philosophical Works', | 1875-90; Routledge & Kegan Paul, London, UK, 1951; | Open Court, La Salle, IL, 1985. Paragraph 360, Page 341. To round out the presentation of the "Polymorphous" Example 1, I will go through what has gone before and lay in the graphic forms of all of the propositional expressions. These graphs, whose official botanical designation makes them out to be a species of "painted and rooted cacti" (PARC's), are not too far from the actual graph-theoretic data-structures that result from parsing the Cactus string expressions, the "painted and rooted cactus expressions" (PARCE's). Finally, I will add a couple of venn diagrams that will serve to illustrate the "difference opus" Dq. If you apply an operator to an operand you must arrive at either an opus or an opera, no? Consider the polymorphous set Q of Example 1 and focus on the central cell, described by the conjunction of logical features in the expression "u v w". o-------------------------------------------------o | X | | | | o-------------o | | / \ | | / \ | | / \ | | / \ | | o U o | | | | | | | | | | | | | | o---o---------o o---------o---o | | / \%%%%%%%%%\ /%%%%%%%%%/ \ | | / \%%%%%%%%%o%%%%%%%%%/ \ | | / \%%%%%%%/%\%%%%%%%/ \ | | / \%%%%%/%%%\%%%%%/ \ | | o o---o-----o---o o | | | |%%%%%| | | | | V |%%%%%| W | | | | |%%%%%| | | | o o%%%%%o o | | \ \%%%/ / | | \ \%/ / | | \ o / | | \ / \ / | | o-------------o o-------------o | | | | | o-------------------------------------------------o Figure 1. Polymorphous Set Q The proposition or truth-function q : X -> B that describes Q is represented by the following graph and text expressions: o-------------------------------------------------o | q | o-------------------------------------------------o | | | u v u w v w | | o o o | | \ | / | | \ | / | | \|/ | | o | | | | | | | | | | | @ | | | o-------------------------------------------------o | (( u v )( u w )( v w )) | o-------------------------------------------------o Conjoining the query that specifies the center cell gives: o-------------------------------------------------o | q.uvw | o-------------------------------------------------o | | | u v u w v w | | o o o | | \ | / | | \ | / | | \|/ | | o | | | | | | | | | | | @ u v w | | | o-------------------------------------------------o | (( u v )( u w )( v w )) u v w | o-------------------------------------------------o And we know the value of the interpretation by whether this last expression issues in a model. Applying the enlargement operator E to the initial proposition q yields: o-------------------------------------------------o | Eq | o-------------------------------------------------o | | | u du v dv u du w dw v dv w dw | | o---o o---o o---o o---o o---o o---o | | \ | | / \ | | / \ | | / | | \ | | / \ | | / \ | | / | | \| |/ \| |/ \| |/ | | o=o o=o o=o | | \ | / | | \ | / | | \ | / | | \ | / | | \ | / | | \ | / | | \ | / | | \ | / | | \ | / | | \ | / | | \|/ | | o | | | | | | | | | | | @ | | | o-------------------------------------------------o | | | (( ( u , du ) ( v , dv ) | | )( ( u , du ) ( w , dw ) | | )( ( v , dv ) ( w , dw ) | | )) | | | o-------------------------------------------------o Conjoining a query on the center cell yields: o-------------------------------------------------o | Eq.uvw | o-------------------------------------------------o | | | u du v dv u du w dw v dv w dw | | o---o o---o o---o o---o o---o o---o | | \ | | / \ | | / \ | | / | | \ | | / \ | | / \ | | / | | \| |/ \| |/ \| |/ | | o=o o=o o=o | | \ | / | | \ | / | | \ | / | | \ | / | | \ | / | | \ | / | | \ | / | | \ | / | | \ | / | | \ | / | | \|/ | | o | | | | | | | | | | | @ u v w | | | o-------------------------------------------------o | | | (( ( u , du ) ( v , dv ) | | )( ( u , du ) ( w , dw ) | | )( ( v , dv ) ( w , dw ) | | )) | | | | u v w | | | o-------------------------------------------------o The models of this last expression tell us which combinations of feature changes among the set {du, dv, dw} will take us from our present interpretation, the center cell expressed by "u v w", to a true value under the target proposition (( u v )( u w )( v w )). The result of applying the difference operator D to the initial proposition q, conjoined with a query on the center cell, yields: o-------------------------------------------------o | Dq.uvw | o-------------------------------------------------o | | | u du v dv u du w dw v dv w dw | | o---o o---o o---o o---o o---o o---o | | \ | | / \ | | / \ | | / | | \ | | / \ | | / \ | | / | | \| |/ \| |/ \| |/ | | o=o o=o o=o | | \ | / | | \ | / | | \ | / | | \ | / | | \ | / | | \ | / | | \ | / u v u w v w | | \ | / o o o | | \ | / \ | / | | \ | / \ | / | | \|/ \|/ | | o o | | | | | | | | | | | | | | o---------------o | | \ / | | \ / | | \ / | | \ / | | \ / | | \ / | | \ / | | @ u v w | | | o-------------------------------------------------o | | | ( | | (( ( u , du ) ( v , dv ) | | )( ( u , du ) ( w , dw ) | | )( ( v , dv ) ( w , dw ) | | )) | | , | | (( u v | | )( u w | | )( v w | | )) | | ) | | | | u v w | | | o-------------------------------------------------o The models of this last proposition are: 1. u v w du dv dw 2. u v w du dv (dw) 3. u v w du (dv) dw 4. u v w (du) dv dw This tells us that changing any two or more of the features u, v, w will take us from the center cell, as described by the conjunctive expression "u v w", to a cell outside the shaded region for the set Q. o-------------------------------------------------o | X | | | | o-------------o | | / \ | | / U \ | | / \ | | / \ | | o @ o | | | ^ | | | | |dw | | | | | | @ | | o---o---------o o----|----o---o ^ | | / \`````````\ /`````|```/ \ /dw | | / du \`````dw``o``dv``|``/ \/ | | / @<-----\-o<----/+\---->o`/ /\ | | / \`````/`|`\`````/ / \ | | o o---o--|--o---o / o | | | |``|``| / | | | | V |`du``| / W | | | | |` |``| / | | | o o``v``o dv / o | | \ \`o-/------->@ / | | \ \`/ / | | \ o / | | \ / \ / | | o-------------o o-------------o | | | | | o-------------------------------------------------o Figure 3. Effect of the Difference Operator D Acting on a Polymorphous Function q Figure 3 shows one way to picture this kind of a situation, by superimposing the paths of indicated feature changes on the venn diagram of the underlying proposition. Here, the models, or the satisfying interpretations, of the relevant "difference proposition" Dq are marked with "@" signs, and the boundary crossings along each path are marked with the corresponding "differential features" among the collection {du, dv, dw}. In sum, starting from the cell uvw, we have the following four paths: 1. du dv dw => Change u, v, w. 2. du dv (dw) => Change u and v. 3. du (dv) dw => Change u and w. 4. (du) dv dw => Change v and w. Next I will discuss several applications of logical differentials, developing along the way their logical and practical implications.

#### DLOG B • Note 5

We have come to the point of making a connection, at a very primitive level, between propositional logic and the classes of mathematical structures that are employed in mathematical systems theory to model dynamical systems of very general sorts. Here is a flash montage of what has gone before, retrospectively touching on just the highpoints, and highlighting mostly just Figures and Tables, all directed toward the aim of ending up with a novel style of pictorial diagram, one that will serve us well in the future, as I have found it readily adaptable and steadily more trustworthy in my previous investigations, whenever we have to illustrate these very basic sorts of dynamic scenarios to ourselves, to others, to computers. We typically start out with a proposition of interest, for example, the proposition q : X -> B depicted here: o-------------------------------------------------o | q | o-------------------------------------------------o | | | u v u w v w | | o o o | | \ | / | | \ | / | | \|/ | | o | | | | | | | | | | | @ | | | o-------------------------------------------------o | (( u v )( u w )( v w )) | o-------------------------------------------------o The proposition q is properly considered as an "abstract object", in some acceptation of those very bedevilled and egging-on terms, but it enjoys an interpretation as a function of a suitable type, and all we have to do in order to enjoy the utility of this type of representation is to observe a decent respect for what befits. I will skip over the details of how to do this for right now. I started to write them out in full, and it all became even more tedious than my usual standard, and besides, I think that everyone more or less knows how to do this already. Once we have survived the big leap of re-interpreting these abstract names as the names of relatively concrete dimensions of variation, we can begin to lay out all of the familiar sorts of mathematical models and pictorial diagrams that go with these modest dimensions, the functions that can be formed on them, and the transformations that can be entertained among this whole crew. Here is the venn diagram for the proposition q. o-----------------------------------------------------------o | X | | | | o-------------o | | / \ | | / \ | | / \ | | / \ | | / \ | | o o | | | U | | | | | | | | | | | | | | | | | | | o--o----------o o----------o--o | | / \%%%%%%%%%%\ /%%%%%%%%%%/ \ | | / \%%%%%%%%%%o%%%%%%%%%%/ \ | | / \%%%%%%%%/%\%%%%%%%%/ \ | | / \%%%%%%/%%%\%%%%%%/ \ | | / \%%%%/%%%%%\%%%%/ \ | | o o--o-------o--o o | | | |%%%%%%%| | | | | |%%%%%%%| | | | | |%%%%%%%| | | | | V |%%%%%%%| W | | | | |%%%%%%%| | | | o o%%%%%%%o o | | \ \%%%%%/ / | | \ \%%%/ / | | \ \%/ / | | \ o / | | \ / \ / | | o-------------o o-------------o | | | | | o-----------------------------------------------------------o Figure 1. Venn Diagram for the Proposition q By way of excuse, if not yet a full justification, I probably ought to give an account of the reasons why I continue to hang onto these primitive styles of depiction, even though I can hardly recommend that anybody actually try to draw them, at least, not once the number of variables climbs much higher than three or four or five at the utmost. One of the reasons would have to be this: that in the relationship between their continuous aspect and their discrete aspect, venn diagrams constitute a form of "iconic" reminder of a very important fact about all "finite information depictions" (FID's) of the larger world of reality, and that is the hard fact that we deceive ourselves to a degree if we imagine that the lines and the distinctions that we draw in our imagination are all there is to reality, and thus, that as we practice to categorize, we also manage to discretize, and thus, to distort, to reduce, and to truncate the richness of what there is to the poverty of what we can sieve and sift through our senses, or what we can draw in the tangled webs of our own very tenuous and tinctured distinctions. Another common scheme for description and evaluation of a proposition is the so-called "truth table" or the "semantic tableau", for example: Table 2. Truth Table for the Proposition q o---------------o-----------o-----------o-----------o-------o | u v w | u & v | u & w | v & w | q | o---------------o-----------o-----------o-----------o-------o | | | | | | | 0 0 0 | 0 | 0 | 0 | 0 | | | | | | | | 0 0 1 | 0 | 0 | 0 | 0 | | | | | | | | 0 1 0 | 0 | 0 | 0 | 0 | | | | | | | | 0 1 1 | 0 | 0 | 1 | 1 | | | | | | | | 1 0 0 | 0 | 0 | 0 | 0 | | | | | | | | 1 0 1 | 0 | 1 | 0 | 1 | | | | | | | | 1 1 0 | 1 | 0 | 0 | 1 | | | | | | | | 1 1 1 | 1 | 1 | 1 | 1 | | | | | | | o---------------o-----------o-----------o-----------o-------o Reading off the shaded cells of the venn diagram or the rows of the truth table that have a "1" in the q column, we see that the "models", or satisfying interpretations, of the proposition q are the four that can be expressed, in either the "additive" or the "multiplicative" manner, as follows: 1. The points of the space X that are assigned the coordinates: <u, v, w> = <0, 1, 1> or <1, 0, 1> or <1, 1, 0> or <1, 1, 1>. 2. The points of the space X that have the conjunctive descriptions: "(u) v w", "u (v) w", "u v (w)", "u v w", where "(x)" is "not x". The next thing that one typically does is to consider the effects of various "operators" on the proposition of interest, which may be called the "operand" or the "source" proposition, leaving the corresponding terms "opus" or "target" as names for the result. In our initial consideration of the proposition q, we naturally interpret it as a function of the three variables that it wears on its sleeve, as it were, namely, those that we find contained in the basis {u, v, w}. As we begin to regard this proposition from the standpoint of a differential analysis, however, we may need to regard it as "tacitly embedded" in any number of higher dimensional spaces. Just by way of starting out, our immediate interest is with the "first order differential analysis" (FODA), and this requires us to regard all of the propositions in sight as functions of the variables in the first order extended basis, specifically, those in the set {u, v, w, du, dv, dw}. Now this does not change the expression of any proposition, like q, that does not mention the extra variables, only changing how it gets interpreted as a function. A level of interpretive flexibility of this order is very useful, and it is quite common throughout mathematics. In this discussion, I will invoke its application under the name of the "tacit extension" of a proposition to any universe of discourse based on a superset of its original basis.

#### DLOG B • Note 6

I think that we finally have enough of the preliminary set-ups and warm-ups out of the way that we can begin to tackle the differential analysis proper of our sample proposition q = (( u v )( u w )( v w )). When X is the type of space that is generated by {u, v, w}, let dX be the type of space that is generated by (du, dv, dw}, and let X x dX be the type of space that is generated by the extended set of boolean basis elements {u, v, w, du, dv, dw}. For convenience, define a notation "EX" so that EX = X x dX. Even though the differential variables are in some abstract sense no different than other boolean variables, it usually helps to mark their distinctive roles and their differential interpretation by means of the distinguishing domain name "dB". Using these designations of logical spaces, the propositions over them can be assigned both abstract and concrete types. For instance, consider the proposition q<u, v, w>, as before, and then consider its tacit extension q<u, v, w, du, dv, dw>, the latter of which may be indicated more explicitly as "eq". 1. Proposition q is abstractly typed as q : B^3 -> B. Proposition q is concretely typed as q : X -> B. 2. Proposition eq is abstractly typed as eq : B^3 x dB^3 -> B. Proposition eq is concretely typed as eq : X x dX -> B. Succinctly, eq : EX -> B. We now return to our consideration of the effects of various differential operators on propositions. This time around we have enough exact terminology that we shall be able to explain what is actually going on here in a rather more articulate fashion. The first transformation of the source proposition q that we may wish to stop and examine, though it is not unusual to skip right over this stage of analysis, frequently regarding it as a purely intermediary stage, holding scarcely even so much as the passing interest, is the work of the "enlargement" or "shift" operator E. Applying the operator E to the operand proposition q yields: o-------------------------------------------------o | Eq | o-------------------------------------------------o | | | u du v dv u du w dw v dv w dw | | o---o o---o o---o o---o o---o o---o | | \ | | / \ | | / \ | | / | | \ | | / \ | | / \ | | / | | \| |/ \| |/ \| |/ | | o=o o=o o=o | | \ | / | | \ | / | | \ | / | | \ | / | | \ | / | | \ | / | | \ | / | | \ | / | | \ | / | | \ | / | | \|/ | | o | | | | | | | | | | | @ | | | o-------------------------------------------------o | | | (( ( u , du ) ( v , dv ) | | )( ( u , du ) ( w , dw ) | | )( ( v , dv ) ( w , dw ) | | )) | | | o-------------------------------------------------o The enlarged proposition Eq is a minimally interpretable as as a function on the six variables of {u, v, w, du, dv, dw}. In other words, Eq : EX -> B, or Eq : X x dX -> B. Conjoining a query on the center cell, c = uvw, yields: o-------------------------------------------------o | Eq.c | o-------------------------------------------------o | | | u du v dv u du w dw v dv w dw | | o---o o---o o---o o---o o---o o---o | | \ | | / \ | | / \ | | / | | \ | | / \ | | / \ | | / | | \| |/ \| |/ \| |/ | | o=o o=o o=o | | \ | / | | \ | / | | \ | / | | \ | / | | \ | / | | \ | / | | \ | / | | \ | / | | \ | / | | \ | / | | \|/ | | o | | | | | | | | | | | @ u v w | | | o-------------------------------------------------o | | | (( ( u , du ) ( v , dv ) | | )( ( u , du ) ( w , dw ) | | )( ( v , dv ) ( w , dw ) | | )) | | | | u v w | | | o-------------------------------------------------o The models of this last expression tell us which combinations of feature changes among the set {du, dv, dw} will take us from our present interpretation, the center cell expressed by "u v w", to a true value under the given proposition (( u v )( u w )( v w )). The models of Eq.c can be described in the usual ways as follows: 1. The points of the space EX that have the following coordinate descriptions: <u, v, w, du, dv, dw> = <1, 1, 1, 0, 0, 0>, <1, 1, 1, 0, 0, 1>, <1, 1, 1, 0, 1, 0>, <1, 1, 1, 1, 0, 0>. 2. The points of the space EX that have the following conjunctive expressions: u v w (du)(dv)(dw), u v w (du)(dv) dw , u v w (du) dv (dw), u v w du (dv)(dw). In summary, Eq.c informs us that we can get from c to a model of q by making the following changes in our position with respect to u, v, w, to wit, "change none or just one among {u, v, w}". I think that it would be worth our time to diagram the models of the "enlarged" or "shifted" proposition, Eq, at least, the selection of them that we find issuing from the center cell c. Figure 4 is an extended venn diagram for the proposition Eq.c, where the shaded area gives the models of q and the "@" signs mark the terminal points of the requisite feature alterations. o-------------------------------------------------o | X | | | | o-------------o | | / \ | | / \ | | / \ | | / \ | | o U o | | | | | | | | | | | | | | o---o---------o o---------o---o | | / \`````````\ /`````````/ \ | | / \`````dw``o``dv`````/ \ | | / \`@<----/@\---->@`/ \ | | / \`````/`|`\`````/ \ | | o o---o--|--o---o o | | | |``|``| | | | | V |`du``| W | | | | |` |``| | | | o o``v``o o | | \ \`@`/ / | | \ \`/ / | | \ o / | | \ / \ / | | o-------------o o-------------o | | | | | o-------------------------------------------------o Figure 4. Effect of the Enlargement Operator E On the Proposition q, Evaluated at c

#### DLOG B • Note 7

One more piece of notation will save us a few bytes in the length of many of our schematic formulations. Let !X! = {x_1, ..., x_k} be a finite class of variables -- whose names I list, according to the usual custom, without what seems to my semiotic consciousness like the necessary quotation marks around their particular characters, though not without not a little trepidation, or without a worried cognizance that I may be obligated to reinsert them all to their rightful places at a subsequent stage of development -- with regard to which we may now define the following items: 1. The "(first order) differential alphabet", d!X! = {dx_1, ..., dx_k}. 2. The "(first order) extended alphabet", E!X! = !X! |_| d!X!, E!X! = {x_1, ..., x_k, dx_1, ..., dx_k}. Before we continue with the differential analysis of the source proposition q, we need to pause and take another look at just how it shapes up in the light of the extended universe EX, in other words, to examine in utter detail its tacit extension eq. The models of eq in EX can be comprehended as follows: 1. Working in the "summary coefficient" form of representation, if the coordinate list x is a model of q in X, then one can construct a coordinate list ex as a model for eq in EX just by appending any combination of values for the differential variables in d!X!. For example, to focus once again on the center cell c, which happens to be a model of the proposition q in X, one can extend c in eight different ways into EX, and thus get eight models of the tacit extension eq in EX. Though it may seem an utter triviality to write these out, I will do it for the sake of seeing the patterns. The models of eq in EX that are tacit extensions of c: <u, v, w, du, dv, dw> = <1, 1, 1, 0, 0, 0>, <1, 1, 1, 0, 0, 1>, <1, 1, 1, 0, 1, 0>, <1, 1, 1, 0, 1, 1>, <1, 1, 1, 1, 0, 0>, <1, 1, 1, 1, 0, 1>, <1, 1, 1, 1, 1, 0>, <1, 1, 1, 1, 1, 1>. 2. Working in the "conjunctive product" form of representation, if the conjunct symbol x is a model of q in X, then one can construct a conjunct symbol ex as a model for eq in EX just by appending any combination of values for the differential variables in d!X!. The models of eq in EX that are tacit extensions of c: u v w (du)(dv)(dw), u v w (du)(dv) dw , u v w (du) dv (dw), u v w (du) dv dw , u v w du (dv)(dw), u v w du (dv) dw , u v w du dv (dw), u v w du dv dw . In short, eq.c just enumerates all of the possible changes in EX that "derive from", "issue from", or "stem from" the cell c in X. Okay, that was pretty tedious, and I know that it all appears to be totally trivial, which is precisely why we usually just leave it "tacit" in the first place, but hard experience, and a real acquaintance with the confusion that can beset us when we do not render these implicit grounds explicit, have taught me that it will ultimately be necessary to get clear about it, and by this "clear" to say "marked", not merely "transparent".

#### DLOG B • Note 8

Before going on -- in order to keep alive the will to go on! -- it would probably be a good idea to remind ourselves of just why we are going through with this exercise. It is to unify the world of change, for which aspect or regime of the world I occasionally evoke the eponymous figures of Prometheus and Heraclitus, and the world of logic, for which facet or realm of the world I periodically recur to the prototypical shades of Epimetheus and Parmenides, at least, that is, to state it more carefully, to encompass the antics and the escapades of these all too manifestly strife-born twins within the scopes of our thoughts and within the charts of our theories, as it is most likely the only places where ever they will, for the moment and as long as it lasts, be seen or be heard together. With that intermezzo, with all of its echoes of the opening overture, over and done, let us now return to that droller drama, already fast in progress, the differential disentanglements, hopefully toward the end of a grandly enlightening denouement, of the ever-polymorphous Q. The next transformation of the source proposition q, that we are typically aiming to contemplate in the process of carrying out a "differential analysis" of its "dynamic" effects or implications, is the yield of the so-called "difference" or "delta" operator D. The resultant "difference proposition" Dq is defined in terms of the source proposition q and the "shifted proposition" Eq thusly: | Dq = Eq - q = Eq - eq. | | Since "+" and "-" signify the same operation over B, we have: | | Dq = Eq + q = Eq + eq. | | Since "+" = "exclusive-or", RefLog syntax expresses this as: | | Eq q Eq eq | o---o o---o | \ / \ / | Dq = @ = @ | | Dq = ( Eq , q ) = ( Eq , eq ). | | Recall that a k-place bracket "(x_1, x_2, ..., x_k)" | is interpreted (in the "existential interpretation") | to mean "Exactly one of the x_j is false", thus the | two-place bracket is equivalent to the exclusive-or. The result of applying the difference operator D to the source proposition q, conjoined with a query on the center cell c, is: o-------------------------------------------------o | Dq.uvw | o-------------------------------------------------o | | | u du v dv u du w dw v dv w dw | | o---o o---o o---o o---o o---o o---o | | \ | | / \ | | / \ | | / | | \ | | / \ | | / \ | | / | | \| |/ \| |/ \| |/ | | o=o o=o o=o | | \ | / | | \ | / | | \ | / | | \ | / | | \ | / | | \ | / | | \ | / u v u w v w | | \ | / o o o | | \ | / \ | / | | \ | / \ | / | | \|/ \|/ | | o o | | | | | | | | | | | | | | o---------------o | | \ / | | \ / | | \ / | | \ / | | \ / | | \ / | | \ / | | @ u v w | | | o-------------------------------------------------o | | | ( | | (( ( u , du ) ( v , dv ) | | )( ( u , du ) ( w , dw ) | | )( ( v , dv ) ( w , dw ) | | )) | | , | | (( u v | | )( u w | | )( v w | | )) | | ) | | | | u v w | | | o-------------------------------------------------o The models of the difference proposition Dq.uvw are: 1. u v w du dv dw 2. u v w du dv (dw) 3. u v w du (dv) dw 4. u v w (du) dv dw This tells us that changing any two or more of the features u, v, w will take us from the center cell that is marked by the conjunctive expression "uvw", to a cell outside the shaded region for the area Q. o-------------------------------------------------o | X | | | | o-------------o | | / \ | | / U \ | | / \ | | / \ | | o @ o | | | ^ | | | | |dw | | | | | | @ | | o---o---------o o----|----o---o ^ | | / \`````````\ /`````|```/ \ /dw | | / du \`````dw``o``dv``|``/ \/ | | / @<-----\-o<----/+\---->o`/ /\ | | / \`````/`|`\`````/ / \ | | o o---o--|--o---o / o | | | |``|``| / | | | | V |`du``| / W | | | | |` |``| / | | | o o``v``o dv / o | | \ \`o-/------->@ / | | \ \`/ / | | \ o / | | \ / \ / | | o-------------o o-------------o | | | | | o-------------------------------------------------o Figure 3. Effect of the Difference Operator D Acting on a Polymorphous Function q Figure 3 shows one way to picture this kind of a situation, by superimposing the paths of indicated feature changes on the venn diagram of the underlying proposition. Here, the models, or the satisfying interpretations, of the relevant "difference proposition" Dq are marked with "@" signs, and the boundary crossings along each path are marked with the corresponding "differential features" among the collection {du, dv, dw}. In sum, starting from the cell uvw, we have the following four paths: 1. du dv dw = Change u, v, w. 2. du dv (dw) = Change u and v. 3. du (dv) dw = Change u and w. 4. (du) dv dw = Change v and w. That sums up, but rather more carefully, the material that I ran through just a bit too quickly the first time around. Next time, I will begin to develop an alternative style of diagram for depicting these types of differential settings.

#### DLOG B • Note 9

Another way of looking at this situation is by letting the (first order) differential features du, dv, dw be viewed as the features of another universe of discourse, called the "tangent universe to X with respect to the interpretation c" and represented as dX.c. In this setting, Dq.c, the "difference proposition of q at the interpretation c", where c = uvw, is marked by the shaded region in Figure 4. o-----------------------------------------------------------o | dX.c | | | | o-------------o | | / \ | | / \ | | / \ | | / \ | | / \ | | o o | | | dU | | | | | | | | | | | | | | | | | | | o--o----------o o----------o--o | | / \``````````\ /``````````/ \ | | / \````2`````o`````3````/ \ | | / \````````/`\````````/ \ | | / \``````/```\``````/ \ | | / \````/``1``\````/ \ | | o o--o-------o--o o | | | |```````| | | | | |```````| | | | | |```````| | | | | dV |```4```| dW | | | | |```````| | | | o o```````o o | | \ \`````/ / | | \ \```/ / | | \ \`/ / | | \ o / | | \ / \ / | | o-------------o o-------------o | | | | | o-----------------------------------------------------------o Figure 4. Tangent Venn Diagram for Dq.c Taken in the context of the tangent universe to X at c = uvw, written dX.c or dX.uvw, the shaded area of Figure 4 indicates the models of the difference proposition Dq.uvw, specifically: 1. u v w du dv dw 2. u v w du dv (dw) 3. u v w du (dv) dw 4. u v w (du) dv dw

#### DLOG B • Note 10

Sub*Title. There's Gonna Be A Rumble Tonight! From: "Theme One: A Program of Inquiry", Jon Awbrey & Susan Awbrey, August 9, 1989. Example 5. Jets and Sharks The propositional calculus that is based on the boundary operator can be interpreted in a way that resembles the logic of activation states and competition constraints in certain neural network models. One way to do this is by interpreting the blank or unmarked state as the resting state of a neural pool, the bound or marked state as its activated state, and by representing a mutually inhibitory pool of neurons A, B, C in the expression "(A, B, C)". To illustrate this possibility, we transcribe a well-known example from the parallel distributed processing literature (McClelland & Rumelhart, 1988) and work through two of the associated exercises as portrayed in Existential Graph format. File "jas.log". Jets and Sharks Example o-----------------------------------------------------------o | | | (( art ),( al ),( sam ),( clyde ),( mike ), | | ( jim ),( greg ),( john ),( doug ),( lance ), | | ( george ),( pete ),( fred ),( gene ),( ralph ), | | ( phil ),( ike ),( nick ),( don ),( ned ), | | ( karl ),( ken ),( earl ),( rick ),( ol ), | | ( neal ),( dave )) | | | | ( jets , sharks ) | | | | ( jets , | | ( art ),( al ),( sam ),( clyde ),( mike ), | | ( jim ),( greg ),( john ),( doug ),( lance ), | | ( george ),( pete ),( fred ),( gene ),( ralph )) | | | | ( sharks , | | ( phil ),( ike ),( nick ),( don ),( ned ),( karl ), | | ( ken ),( earl ),( rick ),( ol ),( neal ),( dave )) | | | | (( 20's ),( 30's ),( 40's )) | | | | ( 20's , | | ( sam ),( jim ),( greg ),( john ),( lance ), | | ( george ),( pete ),( fred ),( gene ),( ken )) | | | | ( 30's , | | ( al ),( mike ),( doug ),( ralph ),( phil ), | | ( ike ),( nick ),( don ),( ned ),( rick ), | | ( ol ),( neal ),( dave )) | | | | ( 40's , | | ( art ),( clyde ),( karl ),( earl )) | | | | (( junior_high ),( high_school ),( college )) | | | | ( junior_high , | | ( art ),( al ),( clyde ),( mike ),( jim ), | | ( john ),( lance ),( george ),( ralph ),( ike )) | | | | ( high_school , | | ( greg ),( doug ),( pete ),( fred ), | | ( nick ),( karl ),( ken ),( earl ), | | ( rick ),( neal ),( dave )) | | | | ( college , | | ( sam ),( gene ),( phil ),( don ),( ned ),( ol )) | | | | (( single ),( married ),( divorced )) | | | | ( single , | | ( art ),( sam ),( clyde ),( mike ),( doug ), | | ( pete ),( fred ),( gene ),( ralph ),( ike ), | | ( nick ),( ken ),( neal )) | | | | ( married , | | ( al ),( greg ),( john ),( lance ),( phil ), | | ( don ),( ned ),( karl ),( earl ),( ol )) | | | | ( divorced , | | ( jim ),( george ),( rick ),( dave )) | | | | (( bookie ),( burglar ),( pusher )) | | | | ( bookie , | | ( sam ),( clyde ),( mike ),( doug ), | | ( pete ),( ike ),( ned ),( karl ),( neal )) | | | | ( burglar , | | ( al ),( jim ),( john ),( lance ), | | ( george ),( don ),( ken ),( earl ),( rick )) | | | | ( pusher , | | ( art ),( greg ),( fred ),( gene ), | | ( ralph ),( phil ),( nick ),( ol ),( dave )) | | | o-----------------------------------------------------------o We now apply 'Study' to the proposition defining the Jets and Sharks data base. With a query on the name "ken" we obtain the following output, giving all the features associated with Ken: File "ken.sen". Output of Query on "ken" o-----------------------------------------------------------o | | | ken | | sharks | | 20's | | high_school | | single | | burglar | | | o-----------------------------------------------------------o With a query on the two features "college" and "sharks" we obtain the following outline of all features satisfying these constraints: File "cos.sen". Output of Query on "college" and "sharks" o-----------------------------------------------------------o | | | college | | sharks | | 30's | | married | | bookie | | ned | | burglar | | don | | pusher | | phil | | ol | | | o-----------------------------------------------------------o From this we discover that all college Sharks are 30-something and married. Further, we have a complete listing of their names broken down by occupation, as no doubt all of them will be, eventually. Reference. | McClelland, James L. & Rumelhart, David E., |'Explorations in Parallel Distributed Processing: | A Handbook of Models, Programs, and Exercises', | MIT Press, Cambridge, MA, 1988. Those who already know the tune, Be at liberty to sing out of it.

#### DLOG B • Note 11

| "The burden of genius is undeliverable" | From a poster, as I once misread it, | Marlboro, Vermont, c. 1976 How does Cosmo, and by this I mean my pet personification of cosmic order in the universe, not to be too tautologous about it, preserve a memory like that, a goodly fraction of a century later, whether localized to this body that's kept going by this heart, and whether by common assumption still more localized to the spongey fibres of this brain, or not? It strikes me, as it has struck others, that it's terribly unlikely to be stored in persistent patterns of activation, for "activation" and "persistent" are nigh a contradiction in terms, as even the author, Cosmo, of the 'I Ching' knew. But that was then, this is now, so let me try to say it planar.

#### DLOG B • Note 12

I happened on the graphical syntax for propositional calculus that I now call the "cactus language" while exploring the confluence of three streams of thought. There was C.S. Peirce's use of operator variables in logical forms and the operational representations of logical concepts, there was George Spencer Brown's explanation of a variable as the contemplated presence or absence of a constant, and then there was the graph theory and group theory that I had been picking up, bit by bit, since I first encountered them in tandem in Frank Harary's foundations of math course, c. 1970. More on that later, as the memories unthaw, but for the moment I want very much to take care of some long-unfinished business, and give a more detailed explanation of how I used this syntax to represent a popular exercise from the PDP literature of the late 1980's, McClelland's and Rumelhart's "Jets and Sharks". The knowledge base of the case can be expressed as a single proposition. The following display presents it in the corresponding text file format. File "jas.log". Jets and Sharks Example o-----------------------------------------------------------o | | | (( art ),( al ),( sam ),( clyde ),( mike ), | | ( jim ),( greg ),( john ),( doug ),( lance ), | | ( george ),( pete ),( fred ),( gene ),( ralph ), | | ( phil ),( ike ),( nick ),( don ),( ned ), | | ( karl ),( ken ),( earl ),( rick ),( ol ), | | ( neal ),( dave )) | | | | ( jets , sharks ) | | | | ( jets , | | ( art ),( al ),( sam ),( clyde ),( mike ), | | ( jim ),( greg ),( john ),( doug ),( lance ), | | ( george ),( pete ),( fred ),( gene ),( ralph )) | | | | ( sharks , | | ( phil ),( ike ),( nick ),( don ),( ned ),( karl ), | | ( ken ),( earl ),( rick ),( ol ),( neal ),( dave )) | | | | (( 20's ),( 30's ),( 40's )) | | | | ( 20's , | | ( sam ),( jim ),( greg ),( john ),( lance ), | | ( george ),( pete ),( fred ),( gene ),( ken )) | | | | ( 30's , | | ( al ),( mike ),( doug ),( ralph ),( phil ), | | ( ike ),( nick ),( don ),( ned ),( rick ), | | ( ol ),( neal ),( dave )) | | | | ( 40's , | | ( art ),( clyde ),( karl ),( earl )) | | | | (( junior_high ),( high_school ),( college )) | | | | ( junior_high , | | ( art ),( al ),( clyde ),( mike ),( jim ), | | ( john ),( lance ),( george ),( ralph ),( ike )) | | | | ( high_school , | | ( greg ),( doug ),( pete ),( fred ), | | ( nick ),( karl ),( ken ),( earl ), | | ( rick ),( neal ),( dave )) | | | | ( college , | | ( sam ),( gene ),( phil ),( don ),( ned ),( ol )) | | | | (( single ),( married ),( divorced )) | | | | ( single , | | ( art ),( sam ),( clyde ),( mike ),( doug ), | | ( pete ),( fred ),( gene ),( ralph ),( ike ), | | ( nick ),( ken ),( neal )) | | | | ( married , | | ( al ),( greg ),( john ),( lance ),( phil ), | | ( don ),( ned ),( karl ),( earl ),( ol )) | | | | ( divorced , | | ( jim ),( george ),( rick ),( dave )) | | | | (( bookie ),( burglar ),( pusher )) | | | | ( bookie , | | ( sam ),( clyde ),( mike ),( doug ), | | ( pete ),( ike ),( ned ),( karl ),( neal )) | | | | ( burglar , | | ( al ),( jim ),( john ),( lance ), | | ( george ),( don ),( ken ),( earl ),( rick )) | | | | ( pusher , | | ( art ),( greg ),( fred ),( gene ), | | ( ralph ),( phil ),( nick ),( ol ),( dave )) | | | o-----------------------------------------------------------o Let's start with the simplest clause of the conjoint proposition: ( jets , sharks ) Drawn as the corresponding cactus graph, we have: jets sharks o-----o \ / \ / @ According to my earlier, if somewhat sketchy interpretive suggestions, we are supposed to picture a quasi-neural pool that contains a couple of quasi-neural agents or "units", that between the two of them stand for the logical variables "jets" and "sharks", respectively. Further, we imagine these agents to be mutually inhibitory, so that settlement of the dynamic between them achieves equilibrium when just one of the two is "active" or "changing" and the other is "stable" or "enduring".

#### DLOG B • Note 13

We were focussing on a particular figure of syntax, presented here in both graph and string renditions: o-------------------------------------------------o | | | x y | | o-----o | | \ / | | \ / | | @ | | | o-------------------------------------------------o | ( x , y ) | o-------------------------------------------------o In traversing the cactus graph, in this case a cactus of one rooted lobe, one starts at the root, reads off a left parenthesis "(" on the ascent up the left side of the lobe, reads off the variable "x", counts off a comma "," as one transits the interior expanse of the lobe, reads off the variable "y", and then sounds out a right parenthesiss ")" on the descent down the last slope that closes out the clause of this cactus lobe. According to the current story about how the abstract logical situation is embodied in the concrete physical situation, the whole pool of units that corresponds to this expression comes to its resting condition when just one of the two units in {x, y} is resting and the other is charged. We may think of the state of the whole pool as associated with the root node of the cactus, here distinguished by an "amphora" or "at" sign "@", but the root of the cactus is not represented by an individual agent of the system, at least, not yet. We may summarize these facts in tabular form, as shown in Table 5. Simply by way of a common term, let's count a single unit as a "pool of one". Table 5. Dynamics of (x , y) o---------o---------o---------o | x | y | (x , y) | o=========o=========o=========o | charged | charged | charged | o---------o---------o---------o | charged | resting | resting | o---------o---------o---------o | resting | charged | resting | o---------o---------o---------o | resting | resting | charged | o---------o---------o---------o I'm going to let that settle a while.

#### DLOG B • Note 14

Table 5 sums up the facts of the physical situation at equilibrium. If we let B = {note, rest} = {moving, steady} = {charged, resting}, or whatever candidates you pick for the 2-membered set in question, the Table shows a function f : B x B -> B, where f[x, y] = (x , y). Table 5. Dynamics of (x , y) o---------o---------o---------o | x | y | (x , y) | o=========o=========o=========o | charged | charged | charged | o---------o---------o---------o | charged | resting | resting | o---------o---------o---------o | resting | charged | resting | o---------o---------o---------o | resting | resting | charged | o---------o---------o---------o There are two ways that this physical function might be taken to represent a logical function: 1. If we make the identifications: charged = true (= indicated), resting = false (= otherwise), then the physical function f : B x B -> B is tantamount to the logical function that is commonly known as "logical equivalence", or just plain "equality": Table 6. Equality Function o---------o---------o---------o | x | y | (x , y) | o=========o=========o=========o | true | true | true | o---------o---------o---------o | true | false | false | o---------o---------o---------o | false | true | false | o---------o---------o---------o | false | false | true | o---------o---------o---------o 2. If we make the identifications: resting = true (= indicated), charged = false (= otherwise), then the physical function f : B x B -> B is tantamount to the logical function that is commonly known as "logical difference", or "exclusive disjunction": Table 7. Difference Function o---------o---------o---------o | x | y | (x , y) | o=========o=========o=========o | false | false | false | o---------o---------o---------o | false | true | true | o---------o---------o---------o | true | false | true | o---------o---------o---------o | true | true | false | o---------o---------o---------o Although the syntax of the cactus language modifies the syntax of Peirce's graphical formalisms to some extent, the first interpretation corresponds to what he called the "entitative graphs" and the second interpretation corresponds to what he called the "existential graphs". In working through the present example, I have chosen the existential interpretation of cactus expressions, and so the form "(jets , sharks)" is interpreted as saying that everything in the universe of discourse is either a Jet or a Shark, but never both at once.

#### DLOG B • Note 15

Before we tangle with the rest of the Jets and Sharks example, let's look at a cactus expression that's next in the series we just considered, this time a lobe with three variables. For instance, let's analyze the cactus form whose graph and string expressions are shown in the next display. o-------------------------------------------------o | | | x y z | | o--o--o | | \ / | | \ / | | @ | | | o-------------------------------------------------o | (x, y, z) | o-------------------------------------------------o As always in this competitive paradigm, we assume that the units x, y, z are mutually inhibitory, so that the only states that are possible at equilibrium are those with exactly one unit charged and all the rest at rest. Table 8 gives the lobal dynamics of the form (x, y, z). Table 8. Lobal Dynamics of the Form (x, y, z) o-----------o-----------o-----------o-----------o | x | y | z | (x, y, z) | o-----------o-----------o-----------o-----------o | | | | | | charged | charged | charged | charged | | | | | | | charged | charged | resting | charged | | | | | | | charged | resting | charged | charged | | | | | | | charged | resting | resting | resting | | | | | | | resting | charged | charged | charged | | | | | | | resting | charged | resting | resting | | | | | | | resting | resting | charged | resting | | | | | | | resting | resting | resting | charged | | | | | | o-----------o-----------o-----------o-----------o Given B = {charged, resting} the Table presents the appearance of a function f : B x B x B -> B, where f[x, y, z] = (x, y, z). If we make the identifications, charged = false, resting = true, in accord with the so-called "existential" interpretation, then the physical function f : B^3 -> B is tantamount to the logical function that is suggested by the phrase "just 1 of 3 is false". Table 9 is the truth table for the logical function that we get, this time using 0 for false and 1 for true in the customary way. Table 9. Existential Interpretation of (x, y, z) o-----------o-----------o-----------o-----------o | x | y | z | (x, y, z) | o-----------o-----------o-----------o-----------o | | | | 0 0 0 | 0 | | | | | 0 0 1 | 0 | | | | | 0 1 0 | 0 | | | | | 0 1 1 | 1 | | | | | 1 0 0 | 0 | | | | | 1 0 1 | 1 | | | | | 1 1 0 | 1 | | | | | 1 1 1 | 0 | | | | o-----------------------------------o-----------o

#### DLOG B • Note 16

I sometimes refer to the cactus lobe operators in the series (), (x_1), (x_1, x_2), (x_1, x_2, x_3), ..., (x_1, ..., x_k) as "boundary operators" and one of the reasons for this can be seen most easily in the venn diagram for the k-argument boundary operator (x_1, ..., x_k). Figure 10 shows the venn diagram for the 3-fold boundary form (x, y, z). o-----------------------------------------------------------o | U | | | | o-------------o | | / \ | | / \ | | / \ | | / \ | | / \ | | o o | | | X | | | | | | | | | | | | | | | | | | | o--o----------o o----------o--o | | / \%%%%%%%%%%\ /%%%%%%%%%%/ \ | | / \%%%%%%%%%%o%%%%%%%%%%/ \ | | / \%%%%%%%%/ \%%%%%%%%/ \ | | / \%%%%%%/ \%%%%%%/ \ | | / \%%%%/ \%%%%/ \ | | o o--o-------o--o o | | | |%%%%%%%| | | | | |%%%%%%%| | | | | |%%%%%%%| | | | | Y |%%%%%%%| Z | | | | |%%%%%%%| | | | o o%%%%%%%o o | | \ \%%%%%/ / | | \ \%%%/ / | | \ \%/ / | | \ o / | | \ / \ / | | o-------------o o-------------o | | | | | o-----------------------------------------------------------o Figure 10. Venn Diagram for (x, y, z) In this picture, the "oval" (actually, octangular) regions that are customarily said to be "indicated" by the basic propositions x, y, z : B^3 -> B, that is, where the simple arguments x, y, z, respectively, evaluate to true, are marked with the corresponding capital letters X, Y, Z, respectively. The proposition (x, y, z) comes out true in the region that is shaded with per cent signs. Invoking various idioms of general usage, one may refer to this region as the indicated region, truth set, or fibre of truth of the proposition in question. It is useful to consider the truth set of the proposition (x, y, z) in relation to the logical conjunction xyz of its arguments x, y, z. In relation to the central cell indicated by the conjunction xyz, the region indicated by "(x, y, z)" is composed of the "adjacent" or the "bordering" cells. Thus they are the cells that are just across the boundary of the center cell, arrived at by taking all of Leibniz's "minimal changes" from the given point of departure.

#### DLOG B • Note 17

Any cell in a venn diagram has a well-defined set of nearest neighbors, and so we can apply a boundary operator of the appropriate rank to the list of signed features that conjoined would indicate the cell in view. For example, having computed the "boundary", or what is more properly called the "point omitted neighborhood" (PON) of the center cell in a 3-dimensional universe of discourse, what is the PON of the cell that is furthest from it, namely, the "origin cell" indicated as (x)(y)(z)? The region bordering the origin cell, (x)(y)(z), can be computed by placing its three signed conjuncts in a 3-place bracket like (__, __, __), arriving at the cactus expression that is shown below in both graph and string forms. o-------------------------------------------------o | | | x y z | | o o o | | | | | | | o--o--o | | \ / | | \ / | | @ | | | o-------------------------------------------------o | ((x),(y),(z)) | o-------------------------------------------------o Figure 11 shows the venn diagram of this expression, whose meaning is adequately suggested by the phrase "just 1 of 3 is true". o-----------------------------------------------------------o | U | | | | o-------------o | | /```````````````\ | | /`````````````````\ | | /```````````````````\ | | /`````````````````````\ | | /```````````````````````\ | | o`````````````````````````o | | |``````````` X ```````````| | | |`````````````````````````| | | |`````````````````````````| | | |`````````````````````````| | | |`````````````````````````| | | o--o----------o```o----------o--o | | /````\ \`/ /````\ | | /``````\ o /``````\ | | /````````\ / \ /````````\ | | /``````````\ / \ /``````````\ | | /````````````\ / \ /````````````\ | | o``````````````o--o-------o--o``````````````o | | |`````````````````| |`````````````````| | | |`````````````````| |`````````````````| | | |`````````````````| |`````````````````| | | |``````` Y ```````| |`````` Z ````````| | | |`````````````````| |`````````````````| | | o`````````````````o o`````````````````o | | \`````````````````\ /`````````````````/ | | \`````````````````\ /`````````````````/ | | \`````````````````\ /`````````````````/ | | \`````````````````o`````````````````/ | | \```````````````/ \```````````````/ | | o-------------o o-------------o | | | | | o-----------------------------------------------------------o Figure 11. Venn Diagram for ((x),(y),(z))

#### DLOG B • Note 18

Given the foregoing explanation of the k-fold boundary operator, along with its use to express such forms of logical constraints as "just 1 of k is false" and "just 1 of k is true", there will be no trouble interpreting an expression of the following shape from the Jets and Sharks example: (( art ),( al ),( sam ),( clyde ),( mike ), ( jim ),( greg ),( john ),( doug ),( lance ), ( george ),( pete ),( fred ),( gene ),( ralph ), ( phil ),( ike ),( nick ),( don ),( ned ), ( karl ),( ken ),( earl ),( rick ),( ol ), ( neal ),( dave )) This expression says that everything in the universe of discourse is either Art, or Al, or ..., or Neal, or Dave, but never any two of them at once. In effect, I've exploited the circumstance that the universe contains but finitely many ostensible individuals to dedicate its own predicate to each one of them, imposing only the requirement that these predicates must be disjoint and exhaustive. Likewise, each of the following clauses has the effect of partitioning the universe of discourse among the factions or features that are enumerated in the clause in question. ( jets , sharks ) (( 20's ),( 30's ),( 40's )) (( junior_high ),( high_school ),( college )) (( single ),( married ),( divorced )) (( bookie ),( burglar ),( pusher )) We may note in passing that ( x , y ) = ((x),(y)), but a rule of this form holds only in the case of the 2-fold boundary operator.

#### DLOG B • Note 19

Let's collect the various ways of representing the structure of a universe of discourse that is described by the following cactus expressions, verbalized as "just 1 of x, y, z is true". o-------------------------------------------------o | | | x y z | | o o o | | | | | | | o--o--o | | \ / | | \ / | | @ | | | o-------------------------------------------------o | ((x),(y),(z)) | o-------------------------------------------------o Table 12 shows the truth table for the existential interpretation of the cactus formula ((x),(y),(z)). Table 12. Existential Interpretation of ((x),(y),(z)) o-----------o-----------o-----------o-------------o | x | y | z | (x, y, z) | o-----------o-----------o-----------o-------------o | | | | 0 0 0 | 0 | | | | | 0 0 1 | 1 | | | | | 0 1 0 | 1 | | | | | 0 1 1 | 0 | | | | | 1 0 0 | 1 | | | | | 1 0 1 | 0 | | | | | 1 1 0 | 0 | | | | | 1 1 1 | 0 | | | | o-----------------------------------o-------------o Figure 13 shows the same data as a 2-colored 3-cube, coloring a node with a hollow dot (o) for "false" or a star (*) for "true". o-------------------------------------------------o | | | x y z | | o | | /|\ | | / | \ | | / | \ | | / | \ | | / | \ | | / | \ | | / x (y) z \ | | x y (z) o o o (x) y z | | |\ / \ /| | | | \ / \ / | | | | \ / \ / | | | | \ / | | | | / \ / \ | | | | / \ / \ | | | |/ \ / \| | | x (y)(z) * * * (x)(y) z | | \ (x) y (z) / | | \ | / | | \ | / | | \ | / | | \ | / | | \ | / | | \|/ | | o | | (x)(y)(z) | | | o-------------------------------------------------o Figure 14 repeats the venn diagram that we've already seen. o-----------------------------------------------------------o | U | | | | o-------------o | | /```````````````\ | | /`````````````````\ | | /```````````````````\ | | /`````````````````````\ | | /```````````````````````\ | | o`````````````````````````o | | |``````````` X ```````````| | | |`````````````````````````| | | |`````````````````````````| | | |`````````````````````````| | | |`````````````````````````| | | o--o----------o```o----------o--o | | /````\ \`/ /````\ | | /``````\ o /``````\ | | /````````\ / \ /````````\ | | /``````````\ / \ /``````````\ | | /````````````\ / \ /````````````\ | | o``````````````o--o-------o--o``````````````o | | |`````````````````| |`````````````````| | | |`````````````````| |`````````````````| | | |`````````````````| |`````````````````| | | |``````` Y ```````| |`````` Z ````````| | | |`````````````````| |`````````````````| | | o`````````````````o o`````````````````o | | \`````````````````\ /`````````````````/ | | \`````````````````\ /`````````````````/ | | \`````````````````\ /`````````````````/ | | \`````````````````o`````````````````/ | | \```````````````/ \```````````````/ | | o-------------o o-------------o | | | | | o-----------------------------------------------------------o Figure 14. Venn Diagram for ((x),(y),(z)) Figure 15 shows an alternate form of venn diagram for the same proposition, where we collapse to a nullity all of the regions on which the proposition in question evaluates to false. This leaves a structure that partitions the universe into precisely three parts. In mathematics, operations that identify diverse elements are called "quotient operations". In this case, many regions of the universe are being identified with the null set, leaving only this 3-fold partition as the "quotient structure". o-----------------------------------------------------------o | \ / | | \ / | | \ / | | \ / | | \ / | | \ X / | | \ / | | \ / | | \ / | | \ / | | \ / | | \ / | | \ / | | \ / | | o | | | | | | | | | | | | | | Y | Z | | | | | | | | | | | | | | | | | | | | | | | | | | | | o-----------------------------o-----------------------------o Figure 15. Quotient Structure Venn Diagram for ((x),(y),(z))

#### DLOG B • Note 20

Let's now look at the last type of clause that we find in my transcription of the Jets and Sharks data base, for instance, as exemplified by the following couple of lobal expressions: ( jets , ( art ),( al ),( sam ),( clyde ),( mike ), ( jim ),( greg ),( john ),( doug ),( lance ), ( george ),( pete ),( fred ),( gene ),( ralph )) ( sharks , ( phil ),( ike ),( nick ),( don ),( ned ),( karl ), ( ken ),( earl ),( rick ),( ol ),( neal ),( dave )) Each of these clauses exhibits a generic pattern whose logical properties may be studied well enough in the form of the following schematic example. o-------------------------------------------------o | | | y z | | o o | | x | | | | o--o--o | | \ / | | \ / | | @ | | | o-------------------------------------------------o | ( x ,(y),(z)) | o-------------------------------------------------o The proposition (u, v, w) evaluates to true if and only if just one of u, v, w is false. In the same way, the proposition (x,(y),(z)) evaluates to true if and only if exactly one of x, (y), (z) is false. Taking it by cases, let us first suppose that x is true. Then it has to be that just one of (y) or (z) is false, which is tantamount to the proposition ((y),(z)), which is equivalent to the proposition ( y , z ). On the other hand, let us suppose that x is the false one. Then both (y) and (z) must be true, which is to say that y is false and z is false. What we have just said here is that the region where x is true is partitioned into the regions where y and z are true, respectively, while the region where x is false has both y and z false. In other words, we have a "pie-chart" structure, where the genus X is divided into the disjoint and X-haustive couple of species Y and Z. The same analysis applies to the generic form (x, (x_1), ..., (x_k)), specifying a pie-chart with a genus X and the k species X_1, ..., X_k.

### Differential Logic • Series C

#### DLOG C • Note 1

It would be good to summarize, in rough but intuitive terms, the outlook on differential logic that we have reached so far. We have been considering a class of operators on universes of discourse, each of which takes us from considering one universe of discourse, X, to considering a larger universe of discourse, EX. Each of these operators, in general terms having the form F : X -> EX, acts on each proposition p : X -> B of the source universe X to produce a proposition Fp : EX -> B of the target universe EX. The two main operators that we have worked with up to this point are the enlargement operator E : X -> EX and the difference operator D : X -> EX. E and D take a proposition in X, that is, a proposition p : X -> B that is said to be "about" the subject matter of X, and produce the extended propositions Ep, Dp : EX -> B, which may be interpreted as being about specified collections of changes that might occur in X. Here we have need of visual representations, some array of concrete pictures to anchor our more earthy intuitions and to help us keep our wits about us before we try to climb any higher into the ever more rarefied air of abstractions. One good picture comes to us by way of the "field" concept. Given a space X, a "field" of a specified type T over X is formed by assigning to each point of X an object of type T. If that sounds like the same thing as a function from X to the space of things of type T, it is, but it does seems to help to vary the mental pictures and the figures of speech that naturally spring to mind within these fertile fields. In the field picture, a proposition p : X -> B becomes a "scalar" field, that is, a field of values in B, or a "field of true-false indications". Let us take a moment to view an old proposition in this new light, for example, the conjunction uv : X -> B that is depicted in Figure 1. o-------------------------------------------------o | X | | | | o-------------o o-------------o | | / \ / \ | | / o \ | | / /`\ \ | | / /```\ \ | | o o`````o o | | | |`````| | | | | U |`````| V | | | | |`````| | | | o o`````o o | | \ \```/ / | | \ \`/ / | | \ o / | | \ / \ / | | o-------------o o-------------o | | | | | o-------------------------------------------------o | f = u v | o-------------------------------------------------o Figure 1. Conjunction uv : X -> B Each of the operators E, D : X -> EX takes us from considering propositions p : X -> B, here viewed as "scalar fields" over X, to considering the corresponding "differential fields" over X, analogous to what are usually called "vector fields" over X. The structure of these differential fields can be described this way. To each point of X there is attached an object of the following type, a proposition about changes in X, that is, a proposition g : dX -> B. In this setting, if X is the universe that is generated by the set of coordinate propositions {u, v}, then dX is the differential universe that is generated by the set of differential propositions {du, dv}. These differential propositions may be interpreted as indicating "change in u" and "change in v", respectively. A differential operator F, of the first order sort that we have been considering, takes a proposition p : X -> B and gives back a differential proposition Fp : EX -> B. In the field view, we see the proposition p : X -> B as a scalar field and we see the differential proposition Fp : EX -> B as a vector field, specifically, a field of propositions about contemplated changes in X. The field of changes produced by E on uv is shown in Figure 2. o-------------------------------------------------o | X | | | | o-------------o o-------------o | | / \ / \ | | / U o V \ | | / /`\ \ | | / /```\ \ | | o o.->-.o o | | | u(v)(du)dv |`\`/`| (u)v du(dv) | | | | o---------------|->o<-|---------------o | | | | |``^``| | | | o o``|``o o | | \ \`|`/ / | | \ \|/ / | | \ o / | | \ /|\ / | | o-------------o | o-------------o | | | | | | | | | | | o | | (u)(v) du dv | | | o-------------------------------------------------o | f = u v | o-------------------------------------------------o | | | Ef = u v (du)(dv) | | | | + u (v) (du) dv | | | | + (u) v du (dv) | | | | + (u)(v) du dv | | | o-------------------------------------------------o Figure 2. Enlargement E[uv] : EX -> B The differential field E[uv] specifies the changes that need to be made from each point of X in order to reach one of the models of the proposition uv, that is, in order to satisfy the proposition uv. The field of changes produced by D on uv is shown in Figure 3. o-------------------------------------------------o | X | | | | o-------------o o-------------o | | / \ / \ | | / U o V \ | | / /`\ \ | | / /```\ \ | | o o`````o o | | | (du)dv |`````| du(dv) | | | | o<--------------|->o<-|-------------->o | | | | |``^``| | | | o o``|``o o | | \ \`|`/ / | | \ \|/ / | | \ o / | | \ /|\ / | | o-------------o | o-------------o | | | | | | | | v | | o | | du dv | | | o-------------------------------------------------o | f = u v | o-------------------------------------------------o | | | Df = u v ((du)(dv)) | | | | + u (v) (du) dv | | | | + (u) v du (dv) | | | | + (u)(v) du dv | | | o-------------------------------------------------o Figure 3. Difference D[uv] : EX -> B The differential field D[uv] specifies the changes that need to be made from each point of X in order to change the value of the proposition uv.

### Differential Logic • Series A • Discussion

#### DLOG A • Discussion Note 1

GR = Gary Richmond JA = Jon Awbrey JS = John Sowa Re: DLOG A10. http://suo.ieee.org/ontology/msg05373.html In Texas they tell a variety of joke that goes a bit like this: | Q. What do you do when your 100-dollar 10-gallon hat | blows off in a dust-storm? | | A. Reach up in the air and pull down another one. The story that I called a "genealogy" not a "history" has to do with the sorts of ideas that are always in the air and only occasionally get seized on in novel fashions. Or call it a Hegelian history if you like. It is customary to give Bentham first billing for the "paraphrasis" idea, so I went along with that. Quine's comment in Van Heijenoort gives additional snippets about Schonfinkel's "Bausteine". A very enjoyable way to learn about combinator calculus is provided by the second half of Ray Smullyan's 'To Mock a Mockingbird'. Folks usually give Curry and Church credit for independently rediscovering what are more or less computationally equivalent ideas in the various lambda calculi. I tend to be suspicious of how independent anybody can be from their collective unconscious background/culture, but that's just me. B. and/or C. Peirce still get credit, I haven't been able to sort out which deserves the lion's share yet, for a very general form of algebraic representation principle that's been a blowin' in the same wind more or less since about the days of Galois. Somewhere in the mid 1970's I figured out the relationship between the pragmatic maxim and these representation principles, and that has been the important thing to me since then. Mathematics and physics are still just about the only places where something like the prag-maxim gets applied on a routine basis, most often by people who never heard of it under that name. I see John's note covers most of the other questions. CSP: | 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. JA: 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. JA: 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: JA: 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. JA: 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, page 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. GR: I managed to follow this discussion pretty well, except for your last comment. JA: 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. GR: Confronted with *lambda calculus*, I think first of what John Sowa has written (see, especially, 'Conceptual Structures' [1984], 162-3, 373-4). A few matters I'm quite unclear about: GR: 1. Sowa describes the lambda calculus in this way (I'm starting in media res, with comments involving a shipping example): GR, quoting JS: JS: | The denotations operation [delta] would search the database to find | some part x, date y, and shipment z that would make the predicates | in the body of the formula true. If the denotation is true, then | the answer to the question is yes. If the denotation is false, | then the answer is no. | | For a wh-type type question *What suppliers shipped parts to Dept. 85?*, | the answer is a set of suppliers. In symbolic logic, a yes-no question | corresponds to a proposition where every variable is *bound* by a quantifier. | A wh-question, however, is mapped to a *lambda expressions* with one or more | parameters ... 162 [the text diagrams this situation through symbolic logic | and a conceptual graph] 162-3 GR: The upshot of this is: JS: | The denotation of the lambda expression is not a truth value like | true or false, but rather the set of all instances of SUPPLIERs that | could be substituted for w to make the body of the expression true. | Lambda calculus combined with symbolic logic makes a powerful database | query language. 163 GR: What do you think of this description, which really constitutes a kind of definition? Is it yours? Is it pointing to what your last, perhaps rhetorical, question is meant to point to? (Collateral knowledge needed, Jon. Less literary device, more information please ;-) GR: 2. In the second passage discussing the lambda calculus there is no mention made of Schönfinkel in Sowa's discussion of the history of the lambda calculus. GR, quoting JS: JS: | A definition by extension is only possible when the domain A is finite. | In all other cases, the function must be defined by a rule, which is | called the *intension* of f. (One could, of course, define the | extension of a function as an infinite set, but the set itself | would have to be defined by some rule or intension.) | | Defining a function by a rule is more natural or intuitive than defining | it as a set of ordered pairs. But a problem can occur when two or more | different rules or intensions lead to the same sets of ordered pairs or | extensions. Are two functions considered the same if they have the same | sets of ordered pairs, but different mapping rules? To distinguish the | intension and extension of functions and to formalize the rules for | defining them, Alonzo Church (1941) developed a system called the | *lambda calculus* which uses the Greek letter [lambda] to indicate | the parameters of a function 373] GR: So what's the exact intellectual history here? I it connected to Peirce in any way? John's earlier discussion hints at this being a triadic logic in the Peircean sense (yes/no being insufficient for the particular purpose at hand -- see the shipping example above) though I don't believe he explicitly states it as such. Later he comments: JS: | An important advantage of the lambda notation is that it defines | a mapping independently of the act of naming it. As a result, an | unnamed lambda expression can be used anywhere that a function name | could be used. This feature is especially useful for applications | that create new functions dynamically and then immediately pass them | as arguments to another function. [e.g., some database systems] GR: John concludes this analysis thus: JS: An important result of the lambda calculus is the Church-Rosser theorem: when more than one function in an expression is expandable, the order of expansion is irrelevant because the same canonical form would be obtained with any sequence of expansion. [He then points to Sect. 3.6 of his text relating this to graphs] 374 GR: What do you think about all this (perhaps even beyond databases)? What are the pragmatic import of the lambda calculus? [Btw, I have not yet read much of John's most recent textbook, so perhaps he's commented on some of these matters there.] GR: If my questions appear naive, I hope you will realize that I am a mere beginning student of logic hoping to get some light thrown on matters that look to me to be of potential (pragmatic) value. GR: Can Peirce's Gamma graphs be related to any of this, btw?

#### DLOG A • Discussion Note 2

RM = Richard Martin Re: DLOG A10. http://suo.ieee.org/ontology/msg05373.html CSP: | 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 Pragmatism", | 'Collected Papers', CP 5.438. RM: I've been pondering this maxim again. If I paraphrase it thus, "My concept of an object's effects is my concept of the object", am I even close to capturing CSP's intent? Even simpler, "My concept of an object is my comprehension of its effects". Am I missing something more profound? The first sentence of the maxim seems to ask no more than to cast "effects" in the broadest of terms. It seems that both of the list servers/archivers that I'm using for this are having problems at the moment, and I have to run out in a second and do some errands, so I will just send out this quick echo now, as a test, and try to get back to your questions later today.

#### DLOG A • Discussion Note 3

RM = Richard Martin Re: DLOG A10. http://suo.ieee.org/ontology/msg05373.html CSP: | 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 Pragmatism", | 'Collected Papers', CP 5.438. RM: I've been pondering this maxim again. If I paraphrase it thus, "My concept of an object's effects is my concept of the object", am I even close to capturing CSP's intent? Even simpler, "My concept of an object is my comprehension of its effects". Am I missing something more profound? The first sentence of the maxim seems to ask no more than to cast "effects" in the broadest of terms. Let me sum up my own rough sense of what I think Peirce is trying to do here. He is trying to identify the critical difference between methods of inquiry that work and those that do not, between those that get us somewhere down the road to greater knowledge and those that just seem to go in circles forever. Experience and history provide lots of examples of both types, always provisionally classified, of course, but there has always been a problem about telling the effective ingredients from the excipients. The maxim expresses one of Peirce's best guesses about what makes the difference, stating it in the form of a practical heuristic that is meant to serve as a guide under all the perplexities of realistic field conditions. Part of the utility of a practical heuristic, a rule of thumb, depends on it implementing a proportionate type of continuity, where using an approximation to the exact rule on a sample of the conceivable data will be proportionately satisficing and not just utterly useless or totally misleading. Robustness, I think they'd call it today. I think what you said above makes a good first approximation, one that I frequently use myself, and one that will serve as a sufficient guideline in the great majority of applications. The way I read it, the Pragmatic Maxim describes and recommends a certain technique for reflecting on, critically examining, and thereby facilitating the clarification and continuous development of one's concepts. At first strike, I see a rule that reminds me of a closure principle. Ordinarily, a closure operator C satisfies a law like C(C(X)) = C(X). But here we have a principle that has the rough shape C(E(X)) = C(X). In words: My conception of the effects of X is my conception of X. Said so succinctly, though, it leaves out a lot of important details, and it's almost impossible to figure out what good such a purported platitude would be unless you look at concrete examples of its use. I started one, but it got more involved than I can track this time of the night, so I will leave it to the morning light. P.S. I'm posting this series to the Inquiry List also, but there's some kind of echo in the Ontology channel, which forced me to isolate it from my other CC lists. Also, the Ontology server is taking about 12 hours to distribute posts, so I will copy you directly.

#### DLOG A • Discussion Note 4

In trying to work through this version of the Pragmatic Maxim one more time, I begin to understand another one of those problems that most of us have in reading Peirce. As a person who read Kant's 'Critique of Pure Reason' up, down, sideways, and, yes, even forward at the age of 13, Peirce sometimes writes as if everybody else has too. Whereas a person like me, who has made two stabs at the buch a decade apart, is still only two thirds of the way through it. Still, I can dimly grasp that many of the things Peirce doesn't say in his statement he doesn't say because he thinks that Kant already painted the background in illustratively enough. One of the bits of fading scenery that I just now noticed is a distinctive ambiguity of force or mood in the use of any maxim for reflective critique, that is, with the aim to bring deliberate reflection to train on a present condition but also to promote a beneficial change of that condition in the future, and thus the effective value of a maxim compounds both descriptive and normative elements. In the present case, the Pragmatic Maxim says something about how to recognize our present concept of an object when we see it, but also how to improve on it. Putting those generalities back in the background, though not entirely wishing to forget them, I'll try once again to get down to details next time.

#### DLOG A • Discussion Note 5

| 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 Pragmatism", | 'Collected Papers', CP 5.438. Let me see what happens if I try to follow Peirce's instructions in the sort of context where they were meant to be used, that is, to clarify the meaning of a concept. Interpreter J picks a concept y whose meaning J wants to clarify. For the sake of the exercise, let us say that y has some object x. We can safely suppose this without making any real commitments of the ontological sort, since we can always abandon the supposition if it leads to absurd consequences, whether logical or practical. As far as what a concept is, Peirce follows a classical tradition that regards a concept as a mental sign. More exactly, a concept is a mental symbol. Now, a symbol is a type of sign that denotes whatever objects it does just because some interpreter interprets it as doing so. Let's put off saying what that means until later, but it is bound to involve Peirce's definition of a sign relation, and so knowing that much allows us to draw a picture of this sort: y o / / x o--------@ \ \ o z This places the 3-ple <x, y, z> of the form <object, sign, sign'> in the context of a suitable sign relation L c !O! x !S! x !I!, where !O!, !S!, and !I! are the relational domains of objects, signs, and interpretant signs available to the interpreter J. Regarding y as a sign, J can ask whether y has objects, and what its objects might be. Regarding y as a symbol, what objects it has depends on its interpreters, for instance, J. That is, it depends on its interpreters in a way that is "essential" and not eliminatable, not without loss of generativity as a symbol. For any interpreter K, let C_K (x) be K's concept of x. For instance, in the present case, we have C_J (x) = y. Relative to the object x of the concept y, the maxim advises J to consider the set of effects, that might conceivably have practical bearings, that J conceives the object x to have. Let E_J (x) be this collection of effects, that might conceivably have practical bearings, that J conceives x to have. The entities and relationships that we've seen so far may be sketched in the form of the following diagram: o y = C_J (x) / / x o--------@ . \ . \ . o z . . . . o C_J (E_J (x)) . / . / o--------@ E_J (x) \ \ o For the sake of a first approximation to the maxim, I am overlooking the subtleties that may be lurking in the proviso "conceivably have practical bearing". Apart from the catches of that one remaining wrinkle, the maxim apparently suggests some sort of descriptive or normative equation between C_J (x) and C_J (E_J (x)). Now, what possible use could such a formula have, when it comes to revealing the meaning of C_J (x)? One thing that comes to mind right off is the similarity of the diagram that I drew above to the kinds that I commonly draw in cases of functions defined by recursion. What I have in mind here is the type of function whose value on "complicated" arguments is arrived at by way of a specified relation to its values on "simpler" arguments. If this form of analogy is apt, then C_J would be analogous to the recursive function in question, and E_J would be analogous to the recursion relation between complex and simple arguments. Of course, it has to be kept in mind that we are talking about structures more general than functions, namely, sign relations. But I think that I will sleep on it now and explore this idea further tomorrow.

#### DLOG A • Discussion Note 6

| 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 Pragmatism", | 'Collected Papers', CP 5.438. We left off last time with this picture: o y = C_J (x) / / x o--------@ . \ . \ . o z . . . . o C_J (E_J (x)) . / . / o--------@ E_J (x) \ \ o We have the following legend for the labels: C_J (x) = J's concept of the object x. E_J (x) = the [set of] effects, that might conceivably have practical bearings, that J conceives x to have. C_J (E_J (x))) = J's concept of the [set of] effects, that might conceivably have practical bearings, that J conceives x to have. I have flagged my intrusion of set-theoretic concepts into Peirce's statement of the Pragmatic Maxim, because there is a potential distortion at this point that we may have to track back to at a later stage if we find that something has gone seriously awry the attempted interpretation. And now that I've stopped to look more carefully at the possible road-blocks, there is what may be a related reservation about Peirce's use of the operator "whole of", as paraphrased in the form: C_J (E_J (x)) is the whole of C_J (x). Is this really meant to be an exact equation, as I've been reading it so far, or is there a more significant operation of integration or synthesis that is being invoked by the 2-adic relative term "whole of"? Let's leave those worries, duly flagged, aside for the moment, and proceed with our attempt at the simplest possible reading.

#### DLOG A • Discussion Note 7

| 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 Pragmatism", | 'Collected Papers', CP 5.438. Re: DLOG A9. http://suo.ieee.org/ontology/msg05372.html In: DLOG A. http://suo.ieee.org/ontology/thrd1.html#05359 Back to the picture that we had last time: o y = C_J (x) / / x o--------@ . \ . \ . o z . . . . o C_J (E_J (x)) . / . / o--------@ E_J (x) \ \ o If my "recursive interpretation of the maxim of pragmatic thinking" (RIOTMOPT) is suited to work out, then it would have to be the case that the "effects" of an object are somehow simpler than the object itself, and simpler in the sense that their concept is easier to conceive than the concept of the object itself. After all, that is just the sort of recourse that one has in recursion, namely, that one's attempt to answer a harder question has "recourse" to the resources of one's answers to easier but related questions. It seems that we must inquire further into the precise nature of these effects, of the sort that "might conceivably have practical bearings". Introducing a variant formulation of the pragmatic maxim, Peirce puts a gloss on the theme of "practical bearings": | Such reasonings and all reasonings turn upon the idea that if one exerts | certain kinds of volition, one will undergo in return certain compulsory | perceptions. Now this sort of consideration, namely, that certain lines | of conduct will entail certain kinds of inevitable experiences is what | is called a "practical consideration". Hence is justified the maxim, | belief in which constitutes pragmatism; namely, | | In order to ascertain the meaning of an intellectual conception one should | consider what practical consequences might conceivably result by necessity | from the truth of that conception; and the sum of these consequences will | constitute the entire meaning of the conception. | | C.S. Peirce, "Pragmatism" (c. 1905), 'Collected Papers', CP 5.9 Very roughly, then, let's try to formalize "effects" as referring to ordered pairs of the form <Volition, Perception>, one might even say, to invert a 2-gone paradigm, pairs of the shape <Response, Stimulus>, Taking up Peirce's more developed description of the subject matter, I will attempt to represent a practical consideration of effects, or an effect that an agent conceives to have practical bearings, as a comprehensive connection between a domain of conduct and a domain of experience, or even just an extensive collection of ordered pairs between certain "lines of conduct" (LOC's) and certain "kinds of inevitable experiences" (KOIE's).

#### DLOG A • Discussion Note 8

Perforce a necessity that custom deems practical, empirical researchers and not just applied mathematicians are accustomed to approach the more refractory objects of inquiry by strategic orders of approximation, and thus to lay down the successive strata of a representation that renders every object, that may be simple enough in its own right, complex to us. We come to the question of whether the analysis is terminable, and thus convergent to a definite result, or interminable, and thus inconclusive in its indications of the object. It is in the setting out of analytic representational series that we see the importance of closure operators, because they clue us in to how an otherwise infinite representation may wrap up in a finite term. Here is one way to see this. A typical form of analytic expansion will generally be conducted with respect to an operator Q in such a way that the successive levels of analysis correlate with increasing powers of Q, as Q^0, Q^1, Q^2, Q^3, ..., and so on. If the operator Q is subject to a law that makes all higher powers redundant after some point, then one has the power to sum up what is logically an infinite series in what is computably a finite term. Some of the simpler operator laws that might turn up, at least, among those that are not entirely trivial, are those of the forms: Q^2 = 0, Q^2 = 1, Q^2 = Q. A "closure operator" C is one that obeys a rule of the last mentioned shape, since requiring C(C(x)) = C(x) for all x is the same as saying that C^2 = C. In algebraic language, one refers to such an operator as being "idempotent". Cf. Kelley, 'General Topology'. http://suo.ieee.org/ontology/msg03874.html I'll discuss this further, later on, in the context of concrete examples.

#### DLOG A • Discussion Note 9

HT = Hugh Trenchard Re: DLOG A Discussion 7. http://suo.ieee.org/ontology/msg05402.html HT: Hi Jon. As usual I can supply only a superficial comment rather than a really rigorous analysis of your discussion. However, I see some parallels here to the concept of emergence, which as we all know is described simplistically as the whole is more than the sum of its parts. The concept actually runs counter to what Peirce said in the quote below, that "the sum of these consequences will constitute the entire meaning of the conception". HT: Even so, I see from your commentary that if your formalism is to work "... the "effects" of an object are somehow simpler than the object itself, and simpler in the sense that their concept is easier to conceive than the concept of the object itself". This appears to me to describe the concept of emergence. HT: I know that you are adhering to a particular line of inquiry, and I'm sure my comments are, more often than not, little more than a distraction to you, but I think it may be a not entirely un-useful observation that what you are doing appears to be consistent (albeit on a very superficial analysis) with other current lines of inquiry, namely regarding descriptions of the emergence of novel forms of order which arise from the interaction of component parts of a complex system. When I find myself using the word "emergence" it is usually to point out some surprising phenomenon that has escaped an adequate explanation by the prevailing frameworks of representation and theory. There are times when it seems like so many people have become so sure of their dominant paradigms that they regard it as some kind of "emergency" whenever one of them is revealed as falling short of reality, but I guess that I've gotten so used to their fallibility that I tend to consider this the normal course of inquiry. In that light, the word "emergence" is sometimes used in a way that bears a number of misleading connotations. Most serious I think is this one: Are we really so sure that the phenomenon has emerged in recent ages, or is it simply that we are just beginning to notice something that has been there all the time? The answer may depend on the case, but it needs to be asked, in any case. So I tend to view the issue this way: There is the reality, and then there are the many different representations that we have of the given reality. It's a convenient figure of speech, but one that we deploy at the risk of self-deception, to say that we analyze an objective reality into its "atoms", "constituents", "elements", "effects", "parts", etc. But any form of analysis that we pick is conducted on the level of our chosen representation, to which we may be very "partial", and not without good reason, but which is always very "partial", and often just wrong, in regard to the objective reality itself. That is why it is critically -- you might even say "Critique-ally" -- important to recall that the concept is a sign, a symbol, a tool whose good is to "grasp", to "seize as one", to "throw together", to "unify a manifold of sense impressions", as the various Greek, Latin, and Anglo-Saxon-Teutonic etymologies will inform us if we stop to examine them closely enough. Remembering this should be enough to remind us that neither the analysis nor the synthesis of the signs that we bring to bear on the case will necessarily touch the real constitution of the object, the phenomenon, or the reality itself. That does not render the representation useless, not by a long shot, it is simply the nature of our representation of reality to function in this partial way.

#### DLOG A • Discussion Note 10

| Such reasonings and all reasonings turn upon the idea that if one exerts | certain kinds of volition, one will undergo in return certain compulsory | perceptions. Now this sort of consideration, namely, that certain lines | of conduct will entail certain kinds of inevitable experiences is what | is called a "practical consideration". Hence is justified the maxim, | belief in which constitutes pragmatism; namely, | | In order to ascertain the meaning of an intellectual conception one should | consider what practical consequences might conceivably result by necessity | from the truth of that conception; and the sum of these consequences will | constitute the entire meaning of the conception. | | C.S. Peirce, "Pragmatism" (c. 1905), 'Collected Papers', CP 5.9 This particular formulation of the pragmatic maxim is personally important to me, as it's the version that linked up my studies in algebra, automata, computer simulation, cybernetics, dynamical systems, and formal languages all through the 1980's to the streams of psychology literature that I was exploring in parallel at the same time. Here is how I was accustomed to picture a "machine with input" in those days: | a b c d | a b c d | a b c d T_1 : | T_2 : | T_3 : | V c d d b V b a d c V d c d b This parameterized set of transformations is given more compactly in Table 1. Table 1. Finite Transducer, or Machine with Input o---------o---------o---------o---------o---------o | | | | | | | | | | a | b | c | d | | V | | | | | o=========o=========o=========o=========o=========o | | | | | | | T_1 | c | d | d | b | | | | | | | | T_2 | b | a | d | c | | | | | | | | T_3 | d | c | d | b | | | | | | | o---------o---------o---------o---------o---------o I copied this out of Ashby's 'Cybernetics', 4/1, except for replacing his letter "R" with my "T".

#### DLOG A • Discussion Note 11

Cf: Pragmatic Maxim. http://suo.ieee.org/ontology/msg05407.html Taking its analogy to various formal principles of closure, recursion, and representation as rough guides, we have been exploring the implications of the pragmatic maxim, so far giving special attention to these two versions: | 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. | | C.S. Peirce, "Issues of Pragmaticism", CP 5.438, (1878/1905). | Such reasonings and all reasonings turn upon the idea that if one exerts | certain kinds of volition, one will undergo in return certain compulsory | perceptions. Now this sort of consideration, namely, that certain lines | of conduct will entail certain kinds of inevitable experiences is what | is called a "practical consideration". Hence is justified the maxim, | belief in which constitutes pragmatism; namely, | | In order to ascertain the meaning of an intellectual conception one should | consider what practical consequences might conceivably result by necessity | from the truth of that conception; and the sum of these consequences will | constitute the entire meaning of the conception. | | C.S. Peirce, "Pragmatism", CP 5.9, (c. 1905). Among the more glaring differences between these two versions we find the words "sum" and "truth" explicitly figuring in the latter variant. I will argue, in good time, that both of these notions are implicitly contained in the sense of the maxim no matter how it may be expressed, but for the time being I'll merely make a note of the point and stick with my plan to deal with first things first. After several days of relaxed reflection, I think that I can now bring some of the points on Hugh Trenchard's "Emergent Phenomena Tangent" back home to roost. The way I see it, we have a reality, say x, and then we have a representation of that reality, say y. If the representation y is "analytic" or "articulate" in any sense of those words, then it will analyze or articulate the reality x in terms of y's components, say, y_1, y_2, y_3, just for a start. So we have a picture like this: x y o------------->o /|\ / | \ / | \ / | \ o o o y_1 y_2 y_3 One thing that we ought to appreciate at this point is that the components of the representation are things that may or may not correspond in any simple way to components of the reality, even if the reality in question can be said to decompose in some way. Such a direct correspondence is optional for representations in general, and the utility of the representation, broadly treated, is independent of our being able to "project" or to "reify" the parts of y as the pieces of x. Just by way of a grammar school example, think of the knuckle-rapping cautions against reifying the subject-predicate structure of natural language syntax into ontological categories. At least, that was the rule when I was in school, and I remember it well.

#### DLOG A • Discussion Note 12

I return to my picture of the relation between a reality x and a representation y whose components are y_1, y_2, y_3. JA: The way I see it, we have a reality, say x, and then we have a representation of that reality, say y. If the representation y is "analytic" or "articulate" in any sense of those words, then it will analyze or articulate the reality x in terms of y's components, say, for example, y_1, y_2, y_3, just for a start. So we have a picture like this: x y o------------->o /|\ / | \ / | \ / | \ o o o y_1 y_2 y_3 Let me now draw what I consider to be a critical distinction with respect to the question of emergence, that is, to put it in the roughest possible terms, the issue of whether "the whole is more than the sum of its parts". If by "whole" we mean the object reality x, and if by "parts" we mean the parts of speech, so to speak, of the representation y, then the emergence of x beyond the y_j is hardly surprising, indeed, it's a corollary of the fact that the representation y is approximate, and thus it proves nothing about the potential emergence of y beyond its own components on the plane of the given representation. Of course, this issue will only be confused still more by the unreflective reification of representational components. On the representational plane, however, the utility of the representation generally depends on each representation being determined by its components. What results from this requirement of useful representations is an obligation to render more explicit what we mean by "parts" and by "sum" in a given setting. For example, if y is simply the set {y_1, y_2, y_3}, then y is something that can be said to exist "over and above" its elements, at least, in some sense, even in the case of a singleton set, say, z = {z_1}. But any claim of "emergence" for the relationship of a set to its elements would most likely be discounted as trivial, since the set is defined as something that is fully determined by its elements.

#### DLOG A • Discussion Note 13

HT = Hugh Trenchard JA = Jon Awbrey Re: DLOG A Discussion 12. http://suo.ieee.org/ontology/msg05421.html Copied with corrections here: JA: The way I see it, we have a reality, say x, and then we have a representation of that reality, say y. If the representation y is "analytic" or "articulate" in any sense of those words, then it will analyze or articulate the reality x in terms of y's components, say, for example, y_1, y_2, y_3, just for a start. So we have a picture like this: x y o------------->o /|\ / | \ / | \ / | \ o o o y_1 y_2 y_3 JA: Let me now draw what I consider to be a critical distinction with respect to the question of emergence, that is, to put it in the roughest possible terms, the issue of whether "the whole is more than the sum of its parts". JA: If by "whole" we mean the object reality x, and if by "parts" we mean the parts of speech, so to speak, of the representation y, then the emergence of x beyond the y_j is hardly surprising, indeed, it's a corollary of the fact that the representation y is approximate, and thus it proves nothing about the potential emergence of y beyond its own components on the plane of the given representation. Of course, this issue will only be confused still more by the unreflective reification of representational components. JA: On the representational plane, however, the utility of the representation generally depends on each representation being determined by its components. What results from this requirement of useful representations is an obligation to render more explicit what we mean by "parts" and by "sum" in a given setting. For example, if y is simply the set {y_1, y_2, y_3}, then y is something that can be said to exist "over and above" its elements, at least, in some sense, even in the case of a singleton set, say, z = {z_1}. But any claim of "emergence" for the relationship of a set to its elements would most likely be discounted as trivial, since the set is defined as something that is fully determined by its elements. HT responds: HT: I haven't given as much thought to this topic as I might have liked this weekend. However, let me just throw another scenario out for discussion. HT: It looks to me you've described a static scenario in which the components of reality x do not interact with one another -- they are simply properties of reality x. This would be akin to describing an apple as consisting of the following properties: it is roughly spherical, it is red, and it is juicy. One of the things that I'm trying to convey here is just how self-conscious we need to be about the gap between the reality and the many alternative representations. Notice how empty the space under x in my picture is. I have not said anything positive about the components of x, and only emphasized how cautious we have to be about imputing the structure of the representation to the reality. Now it's true that a representation is always kind of static in a way, verbs as words are not more fluid than nouns, and a program is a fixed code even if we run it to simulate some process. But that is a property of representations, and I have not projected any of it onto the unknown x. HT: I'm not clear what you mean by "parts of speech" as being the components which comprise the whole of reality x, but if they are linguistic/semiotic elements which represent properties of reality x, then the apple analogy seems to apply. That was a figure of speech. My intent was to stress the fact that the y_j are parts of y, not parts of x. HT: However, what happens when you complicate the scenario so as to describe components which interact? As things stand, it would be complicating the scenario a bit just to posit components for x, for example, as depicted in the following way: x y o------------->o ... /|\ . . . / | \ . . . / | \ . . . / | \ . . . o o o x_1 x_2 x_3 y_1 y_2 y_3 Now, even if I were so bold to as to risk such a step at this point in the development of my representation, what would that really mean? Fiat Lux? Not my apple. In fact, I never deplaned the plane of representation, much less in a way that would harm any realities with my filmy images. Yes, it is probably a very safe bet that everyone in this theatre has already bought into the "physical symbol system thesis" (PSST), and so we can admit that even our imaginings are bound by surly bounds of earthly constraints, but talking about that is still another matter, logically speaking, from the matter of reality x that I chase in the present frame. So I continue to recommend a great deal of caution with respect to this step, to advise that we consider very carefully just what it means to take it, and to suggest that there may be more ways to take it than are dreamt of til now.

#### DLOG A • Discussion Note 14

HT = Hugh Trenchard JA = Jon Awbrey Re: DLOG A Discussion 12. http://suo.ieee.org/ontology/msg05421.html HT: From some of the discussions you've presented, it seems that Peircean logic does contemplate formalizing interactions between components. That would be a misapprehension on several scores: First, as to what I've been busy about under this subject line, and along the lines of inquiry that I've been tracing in its name for the last decade and a half or so, it is very much about extending our most rudimentary logical formalisms in a way that would be more adequate to dealing with the problems of change and diversity as we run into them in qualitative settings. Second, as to the character of Peirce's "logic as semiotics" (LAS), that I am merely drawing on as the best resource for my own effort. Third, as to the purpose of logic in general, a normative science dedicated to the objective of optimizing the practice of thinking, in other words, that investigates the question of how we may best conduct our thoughts, on the condition that we desire to achieve a number of more or less well-appreciated purposes of reasoning. HT: I am reminded, as I mentioned previously, of your discussion relating to measures of uncertainties and options at succeeding junctures ("what to do, what to hope ..."). Even in the basic scenario you presented, which seems to imply some degree of causal interactivity, I recall that compound uncertainty was multiplicative (though additive by logarithm), which even in that sense begins to look more like "emergence" than simply the whole being the sum of its parts. But beyond that, I wonder if there is potential to combine "trees", if you will, of compound uncertainty, that lead to patterns of compound uncertainty which wereperhaps not predictable by the initial set of "decisions" at the base of the "trees". I continue to detect in this interpretation of what I wrote on that thread some assumptions that I never intended to convey, but I do not know how to revise my story in a way that would fix the evident deficits in my account. Part of the problem seems to come in with my use of trees to represent what is really the stepwise analysis of a present situation, and not necessarily a sequential process of decision-making that happens on the plane of action. It may be that my try at a triple play, stretching to cover all three bases of Kant's 3-fold inquiry "What's true? What's to do? What's to hope?" has strained the limits of my agility for capturing all three in a general idea. Anyway, I can see no way home, nor any way to retire the field at this time. HT: I realize that emergence in the sense I have applied it, occurs primarily in the context of complex dynamical systems, and may simply be inapplicable to Peircean logic. But I do wonder if one might stumble upon the existence of fractals or power laws in the patterns of relatively complex uncertainty trees. I would suggest that if such patterns exist, then there is a fundamental relationship between Peircean logic and complexity (in the sense of the science of complexity). Here I get that talking past each other feeling again. Just about all the disciplines that we lump together under the "formal sciences" are designed to address complexity of one ilk or another, and if there is anything that all of them have in common, it is a high level of care about what it takes to attribute any property at all to an objective reality, as distinguished from the careless attribution of a relative property to one of the parties to the relation in question. Now Peirce is remarkable for his early sense of the subtleties that are involved in such questions, but the insights he foresaw are hardly unique today, indeed, they are prevalent in most of the formal sciences that I have ever encountered. HT: I think I know what the general response is going to be -- but here again, I just throw this out as food for thought. Well, anticipated or not I will go ahead and try to give a succinct summary of how I see the question of emergence. I am not questioning the value of the word in pointing to phenomena that escape explanation in terms of our prevailing "frames of inquiry" (FOI's), indeed, it seems to follow almost automatically from the approximate nature of all human FOI's that there will always be these kinds of emergent complications arising from time to Timbuktu. Given all of the things that we ignore in order to form a FOI in the first place, it's a global no-brainer that some of the details we neglected will rise up and bite us back sooner or later. But it does not address that very acute danger if we use the terms Chaos, Complexity, Emergence, Manifoldness, Multiplicity, Variety, Uncertainty, ..., in ways that consign them to the very long list of mystifying vital principles that function not to urge inquiry but solely to pacify its urgings. It's all about how the words are used in practice, not essentially a question of spelling. The question is the next question, after you've succeeded in calling attention to an emerging phenomenon, then what?

#### DLOG A • Discussion Note 15

RM = Richard Martin RM: On a different plane I am flying. Flight, whether of a bird or airplane, is an often mentioned emergent property. So if I assign flight's reality to x, which assignments to the components of y might yield the emergence of flight? For my piece I'm still building boats in the basement root-cellars of logic, so I'll not be uplifting you with any new theories of aerodynamics. Let's just say that x is the flight of the bumblebee, which not to shine you on appears to demand the emergence of some new height of virtuosity in any medium that can carry it, and is often mentioned as a case where all the king's measures and all the king's scores fail to account for the living actuality of the live performance. I think I catch the drift of that. Now, our personal best theory of how it's done will be transcrypted in partial differential equations, with differential coefficients that blur before my eyes in all their jiffily rushing complexity, and so by the grace of Lethe I'll just hint at the shape and flow of their parody in the conventions of that Grimm Marchen style as y_1, y_2, y_3, and by these three to say we play at summing up the eternal infinity, if more in the breach than in the observables.

#### DLOG A • Discussion Note 16

HT = Hugh Trenchard Re: DLOG A Discussion 14. http://suo.ieee.org/ontology/msg05425.html In: Differential Logic A. http://suo.ieee.org/ontology/thrd1.html#05359 It seems that many of these consternations are accountable to defects of my exposition that I do not know how to remedy right now, but I will try to address some of the less entangled ones. HT: Thanks for the succinct exposition. I think I have understood the whole time that Peircean logic (albeit formalized logic at any level) involves a set of axioms and procedures which are fundamentally abstract and do not necessarily allow for the overlap of other or competing formalisms (although I take your point that Peircean logic is not entirely unique). I do not think that "lack of overlap" follows from abstraction. Indeed, one of the reasons for scrambling up the concrete rocks of experience to some higher level of abstraction is just so we can better see the commonalities among domains that appear to be disparate when our nose is to their grindstones. And one of the reasons that I started this work on differential logic was to get a better focus on the principles from mathematical systems theory that might be analogized and generalized to deal with outstanding issues of change and diversity in logic-based intelligent systems. Now, I really do understand the impatience to get on to the interesting bits. The literature that you mention, or the precursors of it, evokes memories of the very sort of excitements that filled my days in the 70's and 80's of the late great 1900's. But -- famous first words -- that was only the beginning. But there is a prerequisite subject that we should have taken in school in order to sew up this very overlap, to forge the links of necessity between the quantitative-probabilistic and the qualitative-logical, but we can't have taken it because it hadn't been invented yet, and so I've been busy working on that. HT: And I do think that current studies of emergent phenomena involve very particular formalisms -- that it isn't just about how and where we might wish to insert our own interpretation of what we want to call an emergent phenomenon. One only needs to review such articles as "Collective Induced Computation" by Delgado and Sole, or Per Bak or countless others to see how emergent phenomena may be analyzed mathematically. The fact that we are bound to use particular formalisms and systems of concepts to talk and to think about phenomena of any kind does not entail that "all is vanity and subjective". But there are general strategies that all formal sciences use to identify the objective properties that may be lurking under the bushes of appearance. As a very typical example, consider the way that the Chomsky-Schutzenberger complexity of a formal language is defined. A given formal language will have many formal grammars, or, alternatively, many automata, that accept or generate it. Now, formal grammars and automata can be classified by ostensible complexity-like properties that they wear on their sleeves, that is, you can easily identify a context-free grammar or a pushdown automaton simply by inspecting the form of its given presentation. A formal language gets to be called "context-free" if there is 'some' context-free grammar or pushdown automaton that accepts or generates it, but that only gives an upper bound on the complexity of the language. In order for it to be called "properly context-free" one has to prove that there are 'no' finite-state automata or grammars that do the job. This definition of formal language complexity exemplifies a very common pattern for defining invariant properties. If one thinks of a grammar as a "theory" of a language, that is, as a particular way of describing a language in effective terms, then the attribution of a property as an invariant of a language, as with the case of graduated complexity properties, entails the idea that the property in question can be demonstrated to belong to the language in question from the standpoint of every relevant description, grammar, or theory of the language. Extracting the general pattern, the attribution of a property that merits being named an "invariant" or an "objective" property of the object under survey will typically involve a logical quantification, like "all" or "some", ranging over a set of theoretical perspectives.

#### DLOG A • Discussion Note 17

Let me see if I can still remember how it came about that the pragmatic maxim emerged in the context of pouring the foundations for differential logic. I was looking at sets of propositions, conceived as sets of functions of the form f : B^k -> B. Then I contemplated the action of a couple of operators on these sets of propositions, operators that were meant as logical analogues of the usual finite difference operators E and D. Of course, E and D refer to whole parameterized families of operators, where "E" denotes the enlargement or shift operators, and "D" denotes the difference or delta operators. I wrote all this out for the case of k = 2 starting at this point: Re: Differential Logic A6. http://suo.ieee.org/ontology/msg05368.html Table 4. E(f) 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 In this context, the set of shift operators acts as a mathematical group on the set of propositions, splitting it up into disjoint and exhaustive subsets that are technically known as "group-reduced equivalence classes", but that are rather more commonly known as "orbits". Generally speaking, all of the elements in the same orbit have some common property that is said to be "preserved" by the action of the group. For example, consider the case for k = 2 that is presented in Table 4. The set of shift operators, redescribed as a set of 4 transformations, T_ij in {T_00, T_01, T_10, T_11}, forms a mathematical group that acts on the set of 16 propositions, in such a way that the 16 propositions are partitioned into exactly 7 orbits. Looking over the propositions in each orbit we can see that the members of the same orbit all have similar geometric "shapes" when viewed as figures in a venn diagram. Why is this important? Here are some reasons that come to mind: 1. Group invariants afford us with one of the ways that invariant properties of objects, actual and formal, commonly come to be recognized in practice. 2. Group actions, by gathering together "birds of a feather", that is, objects that have isomorphisms between them or that share some order of similar structure, into common orbits, act to reduce the complexity of the underlying domain in ways that may also reduce the computational complexity of working with objects in that domain. 3. Group representations, that is, the representations of groups that we get by considering them as sets of operators on sets of relatively concrete objects, are prime examples of how to form operational definitions of abstruse concepts. In this way they provide us with useful guidance as to how we might apply the pragmatic maxim in more general cases. For example, Peirce gave operational representations of concepts like "truth" and "falsity" by means of the same sort of tactic. Now, you must not expect the last word on defining a difficult concept to come from such an isolated form of representation, but even the smallest exemplar of the concept can make its contribution to dispelling its more problematic mysteries.

#### DLOG A • Discussion Note 18

One of the times -- or is it most of the times? -- that the gap between a reality and a representation becomes rather glaringly obvious is when we have several representations of what is regarded as the same reality, for instance, as illustrated in the following picture for the reality x and the alternative representations u and v. u x v o<-------------o------------->o /|\ /|\ / | \ / | \ / | \ / | \ / | \ / | \ o o o o o o u_1 u_2 u_3 v_1 v_2 v_3 There are times when the variant representations are easily brought into felicitous correspondence with one another, and then there are times when the different views are so incommensurable that we begin to wonder whether they really lookout on the same worldscape at all. In the examples of regular representations of groups that I discussed a while back, we have already seen cases of variant representations of the same object that are easily related to each other at various levels of abstraction. Just for a stock example of alternative representations that are not quite so trivially related to each other, one might think of the complementary interpretations of the same quantum phenomenon in terms of particles and in terms of waves.

#### DLOG A • Discussion Note 19

HT = Hugh Trenchard Re: DLOG A Discussion 14. http://suo.ieee.org/ontology/msg05425.html In: Differential Logic A. http://suo.ieee.org/ontology/thrd1.html#05359 HT: I do also realize that unless I can deliver the formal goods myself on any possible overlap, there isn't much point in my going on about it. HT: My only purpose in raising the questions is to see if perhaps you yourself might see some underlying common mathematical principle which remains to be identified. I see that you do not believe there is one, nor are you about to make finding one an objective of yours, and I certainly don't blame you. I do not see how you see that. HT: In terms of your last question about where does one go once he has identified an emergent phenomenon, one might conceivably do what, for example, Delgado and Sole have done, which is to address the question of whether there is "a tradeoff between the individual complexity and collective behavior in such a way that complex emergent properties cannot appear if individuals are too much complex", after which they proceeded by rigourous statistical analysis to answer the question. HT: Or, do as Per Bak did: identify the properties of emergent phenomena (e.g. self-organized criticality), identify the patterns of order which characterize emergent phenomena like power laws (such as Zipf's law or Pareto's law), or fractals (as I mentioned in a previous email). HT: Or do as Langland, Bak, Wolfram, and others have done in simulating the phenomena by computer analysis to show how complex phenomena does indeed derive from very simple rules. To boot, Wolfram believes practically all of science can be examined in terms of cellular automata (see "A New Kind of Science"). (Maybe I should be asking Steven Wolfram this question (?)). HT: These are simply a few examples. Yes, those are very good examples of what one customarily does next. But I hope you understand that nobody goes about developing rigorous mathematical or vigorous statistical models of any phenomenal domain if they feel compelled to sacrifice their rigor and their vigor to the spectre of emergence every time they try to use a plus sign. HT: In terms of power laws, as we all know, they are ubiquitous in complex dynamic systems (this is what I am doing in my analysis of bicycle racing -- I am trying to confirm the existence of a power law relationship between the number and intensity of "attacks"), and given their relative omni-presence, I don't think it entirely unreasonble to suggest they might appear in surprising ways in other formalisms (if such formalisms describe in some way physical phenomena), as unlikely as it might seem. Although again, unless I can make some sort of intelligent connection myself, there isn't much point in pursuing it. All of the things that you mention here are somewhere on my list of "How I Got Into This" (HIGIT), but the first thing that I discovered when I first got into this, so many blue moons ago, is that a whole lot of the spadework for building those castles in the air just hadn't been done yet, so the next thing for me to do was just to get down tuit. And "There You Were" as Wayne & Schuster say. HT: Having said all that, from this point forward I'll stay inside the Peircean box, unless we jump to a completely separate box should we wish, but never the twain shall meet. Ok, if you view it as a box you should be very careful about getting intuit. I sure know I would. But from our station outside the box, we may reach in and find a few tools or toys that have the goods that tools and toys may do. And if it's a vehicle that can take you somewhere you want to go and cannot get there as quickly on your own power, then sometimes you have to overcome your fear of flying and make what mileage you can. HT: p.s. famous last words? Zoom, Zoom, Zoom

#### DLOG A • Discussion Note 20

| Yea, from the table of my memory | I'll wipe away all trivial fond records, | All saws of books, all forms, all pressures past, | That youth and observation copied there, | And thy commandment all alone shall live | Within the book and volume of my brain | Unmixed with baser matter. | | Hamlet, 1.5.98-104 Let me go back to one of the places where I came in, where I woke up in one of my math psych courses and began to pay attention to something inside that box of our classroom one buzzy b(l)oomy midsommer's day. As adapted from Ashby's 'Cybernetics' 4/1, here 'tis: Table 1. Finite Transducer, or Machine with Input o---------o---------o---------o---------o---------o | | | | | | | | | | a | b | c | d | | V | | | | | o=========o=========o=========o=========o=========o | | | | | | | T_1 | c | d | d | b | | | | | | | | T_2 | b | a | d | c | | | | | | | | T_3 | d | c | d | b | | | | | | | o---------o---------o---------o---------o---------o As spied from the perspective of my 3rd eye, it struck me as a table of 3-tuples, to wit: <1, a, c> <1, b, d> <1, c, d> <1, d, b> <2, a, b> <2, b, a> <2, c, d> <2, d, c> <3, a, d> <3, b, c> <3, c, d> <3, d, b> At first sight, one usually sees the "input parameters", in this example the identifiers in the set {1, 2, 3}, as signals or signlike data, while the states are seen as objective conditions in which the machine resides for moments at a time and transits between on cue from the input signals. That view of the 3-adic relation M under survey casts it with a type M c S x O x O, where S is a sign domain and O is an object domain. Initially, then, the machine M appears to have some sort of dual type in relation to the typical sign relation L c O x S x S. It is obviously worth considering 3-adic relations of that type, along with many others, but the apparent differences here tend to disappear from more general points of view. For example, one may be thinking of the input parameters as relatively direct data about the state of an external object system, with the purpose of the machine being nothing more than to store information about this object system in the record of its states. From a still more general point of view, all three domains may be the state spaces of systems, and here it is possible to think of cases where there would be 1, 2, or 3 different systems related. Consider the possibilities ...

#### DLOG A • Discussion Note 21

HT = Hugh Trenchard Re: DLOG A Discussion 20. http://suo.ieee.org/ontology/msg05432.html HT: I am wondering if by your statement "from a still more general point of view, all three domains may be the state spaces of systems", you are suggesting that there are a number of inherent data configurations or relations within a system that may be said to represent or define its information content (i.e. ones not obvious or necessarily intended). Once you adopt the system-theoretic point of view, then the only realities are states of systems, and you could even say that there's really only one big system of which everything else is either a shadow or a subsystem. In order to talk about the information content of a system, you have to go back to the basic sign-theoretic situation and ask what it means for the state of a sign system S to convey information about an object system O to an interpretant system I. HT: Perhaps that's a trivial summation, but I am wondering further: when running the grid relations you've shown through a machine which results in a specific output, does the output inherently carry the 3-adic relations you're showing? In a sense the output is the entire state of the system at any given time, or whenever somebody says "time's up". But you can always define a projection from that state to another range of values that you consider to be the designated outputs. HT: Again, trivial perhaps, but maybe this underlying relationship could be useful as a ciphering method, or potentially as a method for reducing the complexity of algorithmic input (complexity here in the sense of quantity of rules and descriptions used to represent a specific object, set of objects, or system). Yes, many such journeys are possible. Did you know that you can download a copy of Ashby, well, his book anyway, from the Cybernetica portal: http://pespmc1.vub.ac.be/ASHBBOOK.html HT: Just more grist for the mill. | I could be bounded in a nutshell | and count myself a king of infinite space, | were it not that I have bad dreams. | | Hamlet, 2.2.256-258 Well, I'm going to risk it anyway. Z Z Z . . .

#### DLOG A • Discussion Note 22

| My tables, | My tables -- meet it is I set it down | That one may smile and smile and be a villain. | | Hamlet, 1.5.107-109 | Meet it is -- or is it join? -- | That error and information | Bear our cognate strife | With us in the middle, | As ambits torn from | A singular womb. | | But leave the space | That promises peace, | With wile enough and | The wareness to boot: | 'Twill amend thy selve. | | Jon Awbrey, 18 Feb 2004

#### DLOG A • Discussion Note 23

To facilitate the comparison with sign relations, let's reset our machine table in the form of a relational database for the 3-adic relation M c X x Y x Z, as shown in Table 2. Table 2. Transduction Triples o---------o---------o---------o | X | Y | Z | o---------o---------o---------o | 1 | a | c | | 1 | b | d | | 1 | c | d | | 1 | d | b | o---------o---------o---------o | 2 | a | b | | 2 | b | a | | 2 | c | d | | 2 | d | c | o---------o---------o---------o | 3 | a | d | | 3 | b | c | | 3 | c | d | | 3 | d | b | o---------o---------o---------o Using the same tactic that we used for the regular representations of groups, we can express the "meaning" of each input parameter as a logical sum of its operational effects, where an effect is defined as an ordered pair of states, let's say in the order <ante-op, post-op>, sometimes written as "ante:post". T_1 = a:c + b:d + c:d + d:b T_2 = a:b + b:a + c:d + d:c T_3 = a:d + b:c + c:d + d:b In a typical control system scenario, an agent has the job, given the ability to choose a sequence of input parameters, to take the system from whatever state it happens to be in to a specified state, that we may regard as the goal state. This type of set-up lends itself to a game-theoretic description, where one player puts the system in any state and another player has to pick a sequence of input parameters that will bring it to the designated goal state. Of course, there are many such games.

#### DLOG A • Discussion Note 24

DL = David Letterman HT = Hugh Trenchard JA = Jon Awbrey WS = Wm Shockshafte Sub*ject Line. Automatopoiesis WS: | My tables, | My tables -- meet it is I set it down | That one may smile and smile and be a villain. | | Hamlet, 1.5.107-109 JA: | Meet it is -- or is it join? -- | That error and information | Bear our cognate strife | With us in the middle, | As ambits torn from | A singular womb. | | But leave the space | That promises peace, | With wile enough and | The wareness to boot: | 'Twill amend thy selve. | | Jon Awbrey, 18 Feb 2004 HT: | Cognate Strife | | Consider once a duple, | And thou hast a tuple. | Consider all that, | and a recursion create; | Such sweet paradox | To abandon safe space | To consider all that | For infinite ends | No more fathomable | Than the one safe place | From where it began | | Hugh Trenchard | February 19, 2004 DL: It's only an exhibition; It's not a competition: Please! No wagering!

#### DLOG A • Discussion Note 25

Refreshed from that idyllic interlude, it's back to the grinstone once again, to the "finite state automaton" (FSA) that Table 2 shows as M c X x Y x Z. Table 2. Transduction Triples o---------o---------o---------o | X | Y | Z | o---------o---------o---------o | 1 | a | c | | 1 | b | d | | 1 | c | d | | 1 | d | b | o---------o---------o---------o | 2 | a | b | | 2 | b | a | | 2 | c | d | | 2 | d | c | o---------o---------o---------o | 3 | a | d | | 3 | b | c | | 3 | c | d | | 3 | d | b | o---------o---------o---------o Graphical modes of visualizing FSA's can be extremely useful when the number of states is relatively small. Figure 3 shows one way of depicting the machine M, as an "edge-painted node-labeled digraph". Here, the term "painted", say, with "palette" X, means that each edge is associated with a subset of X, the term "labeled", say, with "label set" Y = Z, means that there is a bijective map between the nodes and the set Y, which is here equal to Z, and the term "digraph" is the same thing as saying "directed graph". When the context is understood, it will be most convenient to refer to such a construction as the "graph" of the machine M, or Graph(M). o-----------------------------------------------------------o | | | o---o | | / \ | | o o | | | a | | | o o | | / \ / \ | | / o---o \ | | / / | \ | | 2 / / | \ 3 | | ^ / | v | | / / | \ | | / v | \ | | / / 2 | \ | | / / | \ | | o---o / | o---o | | / \ / 1 | / \ | | o o----------->-|---------o o | | | b | v 1 | d | | | o o-----------<-|---------o o | | \ / 13 | / \ / | | o---o | / o---o | | \ | / / | | \ | 123 / / | | \ | ^ / | | \ | / / | | ^ | / v | | 3 \ | / / 2 | | \ |/ / | | \ o---o / | | \ / \ / | | o o | | | c | | | o o | | \ / | | o---o | | | o-----------------------------------------------------------o Figure 3. FSA as Edge-Painted Node-Labeled Digraph Looking over the graph, it should tell us the same thing as any of our previous representations of M, for example: T_1 = a:c + b:d + c:d + d:b T_2 = a:b + b:a + c:d + d:c T_3 = a:d + b:c + c:d + d:b In so many words: 1 takes a to c, b and c to d, and d to b. 2 switches a and b, and switches c and d. 3 maps a to d and cycles through d, b, c.

### Differential Logic • Series B • Discussion

#### DLOG B • Discussion Note 1

HT = Hugh Trenchard HT: Just throwing out a thought (as usual): if one of the units, or logical variables is stable, while the other is active, do we see a "radiating tree" something like this: active active active \ | / active--- \ stable /---active / | \ / active \ active active HT: I know you were probably going to answer that question in the next episode, but I wanted to see if I roughly understand the concept. Consider the logical expression that I write as "(x , y)" and graph in the following way: x y o---o \ / @ According to the story that we are imagining about the relationship of the logical syntax to the state of a quasi-physical system, this expresssion describes the set-up of a neural pool with two mutually inhibitory units x and y when it finally reaches equilibrium, which in the brand of model that we have here can only be one of two ways: (1) x is active and y is stable, or (2) x is stable and y is active. I can't remember if I had this in mind at the time, but I know that shortly after I wrote this up for my master's work I recognized that there were models of competition processes that Grossberg and others had considered that have this very sort of "winner take all" behavior at equilibrium. However, there is a different twist here that I was just about to get into.

#### DLOG B • Discussion Note 2

HT = Hugh Trenchard HT: I should have added to my "radial tree" a description that "stable" represented either x or y, and "active" also either represented x or y, but "active" in either case was a variable in the sense that all the "actives" radiating around the "stable" represent dynamic change. The number of actives in my illustration was purely arbitrary and was set up to show that while one of either x or y remained in stable state, around it revolve either x or y in a state of change. HT: So I'm not sure if this was already apparent -- but does my explanation above change how you may have viewed my previous email? Of course I mean it only as conceptual tool for me to understand how one unit remains stable while another is changing. HT: In any event, I think I take your point that your representation already illustrates the point, and is simpler than mine (Occam's razor being instructive). Once upon a time in the kingdom of AI it wasn't considered sporting unless you tried the methods of your own device out on many problems aside from those of your own device. So I took up the dare of the Jets and Sharks example in that spirit, and had to go along with the assumptions that came with the turf. In this (West Side) story, complex propositions are represented by pools of formal neurons that represent the basic logical variables. It doesn't matter all that much what you call the states, so long as you have two distinguishable states. And because one is trying to imagine how a continuous dynamic system can reliably embody or represent a discrete logical proposition, there is plenty of room to play around with alternative models. I was using the word "stable" for a state of dynamic equilibrium, typically a lower state of activation than the other states that are available to agents, and I was using the word "active" for a higher state of activation, typically transient, that there must be exactly one formal neuron occupying, at equilibrium, in these specialized pools of mutually inhibitory units. And even though the system passes through all sorts of intervening states before it finally settles down, it is only the equilibrium state of the neural pool as a whole that counts. Whatever choice of words eventually works best, the connotation that we do want to preserve is the similarity between the agent at rest and the collective of agents in a state of equilibrium, as this is what gives us a notion of logical value that makes sense for both simple and complex propositions, and thus what allows us to combine propositions in the proper logical way.

#### DLOG B • Discussion Note 3

HT = Hugh Trenchard Thanks for waking me from my documental stupors -- I'm beginning to remember why I put off doing this for 15 years -- that estimate of 10^3 words/picture really is a gross under-statement. HT: I'm not clear on at least a couple of points here. Is the "region bordering the origin cell" that part of the Venn diagram where all three circles overlap? If so, is the "point omitted neighbourhood" those regions of the three circles which do not overlap with any of their neighbours? I called that outer region where all the predicates x, y, z are false the "origin" because its coordinates are <0, 0, 0>. So the cells adjacent to the origin are the gravely accented ones in the following venn diagram: o-----------------------------------------------------------o | U | | | | o-------------o | | /```````````````\ | | /`````````````````\ | | /```````````````````\ | | /`````````````````````\ | | /```````````````````````\ | | o`````````````````````````o | | |``````````` X ```````````| | | |`````````````````````````| | | |```````````100```````````| | | |`````````````````````````| | | |`````````````````````````| | | o--o----------o```o----------o--o | | /````\ \`/ /````\ | | /``````\ 110 o 101 /``````\ | | /````````\ / \ /````````\ | | /``````````\ / \ /``````````\ | | /````````````\ / 111 \ /````````````\ | | o``````````````o--o-------o--o``````````````o | | |`````````````````| |`````````````````| | | |`````````````````| |`````````````````| | | |`````````````````| |`````````````````| | | |``````` Y ```````| |`````` Z ````````| | | |`````````````````| |`````````````````| | | o```````010```````o 011 o``````001````````o | | \`````````````````\ /`````````````````/ | | \`````````````````\ /`````````````````/ | | \`````````````````\ /`````````````````/ | | \`````````````````o`````````````````/ | | \```````````````/ \```````````````/ | | o-------------o o-------------o | | | | 000 | | | o-----------------------------------------------------------o Figure 11. Venn Diagram for ((x),(y),(z)) HT: You note that the phrase "just one of three is true" describes the Venn diagram. This is obviously different from the intuitive sense that "there is a part of x and a part of y and a part of z that are common, or shared, among x, y, and z". So if the centre, common region, represents "one of three is true" (if I have that much right, which I might not), does that mean as between y and z, "one of two is true", and as between x and z "one of two is true", and as between x and y "one of two is true"? No, the center cell is the one where all 3 are true, that is, its coordinates are <x, y, z> = <1, 1, 1>. The shaded region that is indicated by ((x),(y),(z)) has the 3 cells whose coordinates are 100, 010, 001. If I had overlapping gels, these would be the pure primary colors, thus, the cells where just 1 of 3 properties applies, a different 1 in each case.

#### DLOG B • Discussion Note 4

HT = Hugh Trenchard Re: http://stderr.org/pipermail/inquiry/2004-February/001223.html HT: Ah -- looks like my initial interpretation was almost precisely backwards! Things are definitely clearer now. So the origin, or 000 is, so to speak, the remainder of the universe of discourse. (For example, in expropriation law, an area of law I was working in for a while -- I am employed as a paralegal -- one speaks of a portion of land expropriated from an owner as the land "taken" and the area still owned by the land owner as the "remainder"). "Adverse possession" !!! Slowly I turned ... Yes, all of those "cells", the undivided regions, are really equal citizens in the universe of discourse, and it's only the particular form of planar projection that gives them such different shapes in the venn diagram. Another popular way to represent three logical dimensions would be in the form of a 3-cube, like so: o-------------------------------------------------o | | | x y z | | o | | /|\ | | / | \ | | / | \ | | / | \ | | / | \ | | / | \ | | / | \ | | x y (z) o x (y) z o (x) y z | | |\ / \ /| | | | \ / \ / | | | | \ / \ / | | | | \ / | | | | / \ / \ | | | | / \ / \ | | | |/ \ / \| | | x (y)(z) o (x) y (z) o (x)(y) z | | \ | / | | \ | / | | \ | / | | \ | / | | \ | / | | \ | / | | \|/ | | o | | (x)(y)(z) | | | o-------------------------------------------------o In this variety of picture, the cells are the nodes of the cube and the propositions are all the different ways of coloring the nodes of the cube in just 2 colors, corresponding to "indicated" and "undicated", or true and false, under the given proposition. In general, for an n-cube, there are 2^n nodes for the singular elements that the logician calls "interpretations", and 2^(2^n) possible 2-colorings that represent the set of propositions. HT: And the common elements of the three overlapping circles is the point where all three are true, and the "petals" not overlapping with anything are the points where one of three is true. Have I got it right now? Yes, that's the ticket. HT: That still leaves my other question though -- if there is a common region where all three are true, then aren't there also three regions where two cells overlap -- which I guess now means "both are true" (rather than one of two is true, which was my previous interpretation)? Here we come to the question that is sometimes described as the distinction between contemplation and conviction, or the difference between considering a proposition and asserting it. This is also bound up with the difference between mention and use. Historically speaking, a whole lot more noise than signal has been emitted on this score. Consider what we are doing when we draw a venn diagram like this: o-----------------------------------------------------------o | U | | | | o-------------o | | /```````````````\ | | /`````````````````\ | | /```````````````````\ | | /`````````````````````\ | | /```````````````````````\ | | o````````````X````````````o | | |`````````````````````````| | | |`````````````````````````| | | |`````````x (y)(z)````````| | | |`````````````````````````| | | |`````````````````````````| | | o--o----------o```o----------o--o | | /````\ \`/ /````\ | | /``````\ x y (z) o x (y) z /``````\ | | /````````\ / \ /````````\ | | /``````````\ / \ /``````````\ | | /````````````\ /x y z\ /````````````\ | | o``````````````o--o-------o--o``````````````o | | |`````````````````| |`````````````````| | | |`````````````````| |`````````````````| | | |````````Y````````| |```````Z`````````| | | |`````````````````|(x)y z |`````````````````| | | |`````````````````| |`````````````````| | | o````(x) y (z)````o o```(x)(y) z``````o | | \`````````````````\ /`````````````````/ | | \`````````````````\ /`````````````````/ | | \`````````````````\ /`````````````````/ | | \`````````````````o`````````````````/ | | \```````````````/ \```````````````/ | | o-------------o o-------------o | | | | (x)(y)(z) | | | o-----------------------------------------------------------o Figure 11. Venn Diagram for ((x),(y),(z)) One thing that we are doing is according additional attention to the gravely accented areas that we have shaded in like so [```]. That is done in order to represent one particular proposition out of the 2^(2^3) = 2^8 = 256 possible propositions that are available to us in this 3-dimensional universe of discourse. Our practical reasons for representing such a proposition may be various. We may just be thinking about it, we may believe that it's true, we may wish that it were true, we may wish to convince somebody else that it's true whether we believe it ourselves or not, and so on. I'll bet you can think of a slew of other cases. The mathematical way of handling the issue is in terms of concepts that are known as "fibers" and "quotients".

#### DLOG B • Discussion Note 5

I know that it's only Thursday, but it feels like it ought to be Friday, so let me take a break and touch base, if lightly in passing, on a couple of issues that occasionally emerge in this framework, even if my better judgment and worsted experience tells me that I will most likely have to continue putting them off until I'm older if not wiser. Put Off 1. Infinite Expressions I wasn't quite sure what Hugh was getting at with that "radiating tree" business, and the mismatch between our character formats makes my reconstruction of his illustrative Figure just a bit hypothetical, but something about the general drift and shape of what he wrote reminds me of the problem area that concerns the meaning of "infinite expressions", or as it's sometimes met, "re-entrant expressions". HT: Just throwing out a thought (as usual): if one of the units, or logical variables is stable, while the other is active, do we see a "radiating tree" something like this: active active active \ | / \ | / active--o stable o--active / | \ / | \ active active active I will just put a flag in this for now, and maybe revisit it later in the day. I didn't say which day ...

#### DLOG B • Discussion Note 6

HT = Hugh Trenchard HT: I have this feeling that you were a teacher or instructor of some kind once. You have the teacher's knack for planting seeds of thought in the minds of his students. I think my "radiating tree" was an illustration of what you'd said about mutually inhibitory units in a neural pool at equilibrium. In my tree "stable" and "active" could be either x or y, but there is only one "stable" because it does not change, while revolving around it is a series of "actives" -- these are "radiating" or revolving only because they are active. HT: Now that may not be an accurate illustration of your point, but you inspired me to look up "re-entrant/infinite expressions" and one article online I found particularly interesting (granted I didn't look at very many!). www.eeng.dcu.ie/~alife/bmcm9401/vern2.pdf HT: The article touches largely on the notion of self-reference in the context of autopoeitic systems. While the article will require substantially more in-depth study (and much of it is simply far beyond my league), some of the logical discussion did have at least some superficial resemblances to my radiating tree. As I say, however, I am hopelessly unable at this time at least to say anything more on it than that. HT: Have you any references that might be a sort of textbook introduction to re-entrant/infinite expressions? Perhaps if I start with something basic, I can progress to a point where I can see if my radiating trees have any connection to the re-entrant/infinite expressions. HT: "The barber shaves only those men who do not shave themselves". This is in that distributed chapter known as "how I got into this ... and may one day get around to again" (HIGIT ... AMODGATA). There's Spencer Brown's 'Laws of Form', Peter Aczel's 'Non-Well-Founded Sets', Barwise & Etchemendy's 'The Liar', Barwise & Moss's 'Vicious Circles', Manes & Arbib's 'Algebraic Approaches to Program Semantics', and then there's Mandelbrot, just to name a few that come to mind right off. These days, I tend to view all of this stuff as being a question -- not so much of loopiness -- but more of the relationship between the finite and the infinite. And what I found is mainly that we are not nearly as good at the finite as we like to imagine, and and so there's been all this work to do on that side of things. Oh, did you see that there's now a discussion forum for NKS? -- http://forum.wolframscience.com/

### Differential Logic • Series B • Work Area

#### DLOG B • Work Area 1

"vague object impressions" (VOI's) or you might say "hints of objects" (HOO's) "vague impressions of objects" (VIOO's) "impressions of vague objects" (IOVO's) ambit, a unit of ambiguous information that's not a bit till it's interpreted. SPONS ACOYA ("a couple of years ago") pro epi note rest rock roll bound blank spike stone active stable mobile static moving steady charged resting working playing resting changing enduring creation devotion creating reciting producing receiving evanescence equilibrium

#### DLOG B • Work Area 2

Put Off 2. What happened to the fabled distinction between individuals and predicates?

### Differential Logic • Series B • Omitted Text

Speaking of initial conditions -- Though, to speak in truth, if come the day, When do such as we, or all the likes of us, Ever truly speak of our initial conditions, Sensitive toward them how ever we may grow? So let me then speak of initial conditions In just the way I wit then I ultimately do, In medias res, in thick midsts of the plot, For my part there is much that begins here: | The most fundamental concept in cybernetics is that of "difference", | either that two things are recognisably different or that one thing | has changed with time. | | Ashby, W. Ross, |'An Introduction to Cybernetics', | Chapman & Hall, London, UK, 1956, | Methuen & Company, London, UK, 1964, | Page 9. The quickest way for me to open up this topic is just to jump in, and so I will cite here my very first outline of it, that formed an appendix to my Master's (in Psych) Thesis Substitute Document. I am hoping that the very simplicity of this will make it useful as an initial invitation, aided by the circumstance that I wrote it in such a way as to form a gentle bridge between the ordinary difference calculus, executed over the reals R or the integers Z, and the logical difference calculus, working over the booleans B. Accordingly, all of the definitions, equations, expressions, and formulas in the following presentation can be read independently of whether you interpret the ostensible names as denoting values in R, or Z, or B.

### Differential Logic • Series D

Differential Logic and Dynamic Systems Author: Jon Awbrey Created: 16 Dec 1993 Relayed: 31 Oct 1994 Revised: 03 Jun 2003 0. Purpose 1. Review and Transition 2. A Functional Conception of Propositional Calculus 2.1. Qualitative Logic and Quantitative Analogy 2.2. Philosophy of Notation: Formal Terms and Flexible Types 2.3. Special Classes of Propositions 2.4. Basis Relativity and Type Ambiguity 2.5. The Analogy Between Real and Boolean Types 2.6. Theory of Control and Control of Theory 2.7. Propositions as Types and Higher Order Types 2.8. Reality at the Threshold of Logic 2.9. Tables of Propositional Forms 3. A Differential Extension of Propositional Calculus 3.1. Differential Propositions: The Qualitative Analogue of Differential Equations 3.2. An Interlude on the Path 3.3. The Extended Universe of Discourse 3.4. Intentional Propositions 3.5. Life on Easy Street 4. Back to the Beginning: Some Exemplary Universes 4.1. A One-Dimensional Universe 4.2. Example 1. A Square Rigging 4.3. Back to the Feature 4.4. Tacit Extensions 4.5. Example 2. Drives and Their Vicissitudes 5. Transformations of Discourse 5.1. Foreshadowing Transformations: Extensions and Projections of Discourse 5.1.1. Extension from 1 to 2 Dimensions 5.1.2. Extension from 2 to 4 Dimensions 5.2. Thematization of Functions: And a Declaration of Independence for Variables 5.2.1. Thematization: Venn Diagrams 5.2.2. Thematization: Truth Tables 5.3. Propositional Transformations 5.3.1. Alias and Alibi Transformations 5.3.2. Transformations of General Type 5.4. Analytic Expansions: Operators and Functors 5.4.1. Operators on Propositions and Transformations 5.4.2. Differential Analysis of Propositions and Transformations 5.4.2.1. The Secant Operator: $E$ 5.4.2.2. The Radius Operator: $e$ 5.4.2.3. The Phantom of the Operators: !h! 5.4.2.4. The Chord Operator: $D$ 5.4.2.5. The Tangent Operator: $T$ 5.5. Transformations of Type B^2 -> B^1 5.5.1. Analytic Expansion of Conjunction 5.5.1.1. Tacit Extension of Conjunction 5.5.1.2. Enlargement Map of Conjunction 5.5.1.3. Digression: Reflection on Use and Mention 5.5.1.4. Difference Map of Conjunction 5.5.1.5. Differential of Conjunction 5.5.1.6. Remainder of Conjunction 5.5.1.7. Summary of Conjunction 5.5.2. Analytic Series: Coordinate Method 5.5.3. Analytic Series: Recap 5.5.4. Terminological Interlude 5.5.5. End of Perfunctory Chatter: Time to Roll the Clip! 5.6. Taking Aim at Higher Dimensional Targets 5.7. Transformations of Type B^2 -> B^2 Epilogue, Enchoiry, Exodus o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D1 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Differential Logic and Dynamic Systems | Stand and unfold yourself. | | 'Hamlet', 1.1.2 Purpose This series of reports develops a differential extension of propositional calculus and applies it to a context of problems arising in dynamic systems. The work pursued here is coordinated with a parallel application that focuses on neural network systems, but the dependencies are arranged to make the present series the main and the more self-contained work, to serve as a conceptual frame and a technical background for the network project. Review and Transition This note continues a previous discussion on the problem of dealing with change and diversity in logic-based intelligent systems. For ease of reference, I begin by summarizing essential material from previous reports. Table 1 outlines the notation that I use for propositional calculus. Explained as briefly as possible, I am using only two basic kinds of truth-functional connectives, both of variable k-ary scope. 1. For the first, I use concatenation as a connective to indicate the logical conjunction of k arguments. 2. For the other, I use a bracket of the form ( , , , ) as a connective which says that exactly one of its k arguments is false. All other truth-functional connectives can be obtained in a very efficient style of representation through combinations of these two forms. This treatment of propositional logic is derived from the work of C.S. Peirce [P1, P2], who gave this approach an extensive development in his graphical systems of predicate, relational, and modal logic [Rob]. More recently, these ideas were revived and supplemented in an alternative interpretation by G. Spencer-Brown [SpB]. Both of these authors used other forms of enclosure where I use parentheses, but the structural topologies of expression and the functional varieties of interpretation are fundamentally the same. While working with expressions solely in propositional calculus, the use of plain parentheses to represent logical connectives is simplest to write and easiest to read for both human and machine parsers. In the present text I preserve this form of expression in tables and set-off displays, but in contexts where parentheses are needed for functional notation I will use a distinctive font for logical operators. [Not available in Ascii.] The briefest expression for logical truth is the empty word, usually denoted by epsilon or lambda in formal languages, where it forms the identity element for concatenation. To make it visible in this text, I denote it by the equivalent expression "(())", or, especially if operating in an algebraic context, by a simple "1". Also when working in an algebraic mode, I use the plus sign "+" for exclusive disjunction. Thus, we may express the following paraphrases of algebraic forms: A + B = (A, B) A + B + C = ((A, B), C) = (A, (B, C)) One should be careful to observe that these last two expressions are not equivalent to the form (A, B, C). Table 1. Syntax & Semantics of a Calculus for Propositional Logic o-------------------o-------------------o-------------------o | Expression | Interpretation | Other Notations | o-------------------o-------------------o-------------------o | " " | True. | 1 | o-------------------o-------------------o-------------------o | () | False. | 0 | o-------------------o-------------------o-------------------o | A | A. | A | o-------------------o-------------------o-------------------o | (A) | Not A. | A' | | | | ~A | o-------------------o-------------------o-------------------o | A B C | A and B and C. | A & B & C | o-------------------o-------------------o-------------------o | ((A)(B)(C)) | A or B or C. | A v B v C | o-------------------o-------------------o-------------------o | (A (B)) | A implies B. | A => B | | | If A then B. | | o-------------------o-------------------o-------------------o | (A, B) | A not equal to B. | A =/= B | | | A exclusive-or B. | A + B | o-------------------o-------------------o-------------------o | ((A, B)) | A is equal to B. | A = B | | | A if & only if B. | A <=> B | o-------------------o-------------------o-------------------o | (A, B, C) | Just one of | A'B C v | | | A, B, C | A B'C v | | | is false. | A B C' | o-------------------o-------------------o-------------------o | ((A),(B),(C)) | Just one of | A B'C' v | | | A, B, C | A'B C' v | | | is true. | A'B'C | | | | | | | Partition all | | | | into A, B, C. | | o-------------------o-------------------o-------------------o | ((A, B), C) | Oddly many of | A + B + C | | (A, (B, C)) | A, B, C | | | | are true. | A B C v | | | | A B'C' v | | | | A'B C' v | | | | A'B'C | o-------------------o-------------------o-------------------o | (Q, (A),(B),(C)) | Partition Q | Q'A'B'C' v | | | into A, B, C. | Q A B'C' v | | | | Q A'B C' v | | | Genus Q comprises | Q A'B'C | | | species A, B, C. | | o-------------------o-------------------o-------------------o NB. The usage that one often sees, of a plus sign "+" to represent inclusive disjunction, and the reference to this operation as "boolean addition", is a misnomer on at least two counts. Boole used the plus sign to represent exclusive disjunction (at any rate, an operation of aggregation restricted in its logical interpretation to cases where the represented sets are disjoint [Boo, 32]), as any mathematician with a sensitivity to the ring and field properties of algebra would do: | The expression x + y seems indeed uninterpretable, | unless it be assumed that the things represented | by x and the things represented by y are entirely | separate; that they embrace no individuals in | common. [Boo, 66]. It was only later that Peirce and Jevons treated inclusive disjunction as a fundamental operation, but these authors, with a respect for the algebraic properties that were already associated with the plus sign, used a variety of other symbols for inclusive disjunction [Sty, 177, 189]. It seems to have been Schroeder who later reassigned the plus sign to inclusive disjunction [Sty, 208]. Additional information, discussion, and references can be found in [Boo] and [Sty, 177-263]. Aside from these historical points, which never really count against a current practice that has gained a life of its own, this usage does have a further disadvantage of cutting or confounding the lines of communication between algebra and logic. For this reason, I am forced to avoid it here. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D2 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | Out of the dimness opposite equals advance . . . . | Always substance and increase, | Always a knit of identity . . . . always distinction . . . . | always a breed of life. | | Walt Whitman, 'Leaves of Grass', [Whi, 28] A Functional Conception of Propositional Calculus In the general case, we start with a set of logical features {a_1, ..., a_n} that represent properties of objects or propositions about the world. In concrete examples the features {a_i} commonly appear as capital letters from an "alphabet" like {A, B, C, ...} or as meaningful words from a linguistic "vocabulary" of codes. This language can be drawn from any sources, whether natural, technical, or artificial in character and interpretation. In the application to dynamic systems we tend to use the letters {x_1, ... , x_n} as our coordinate propositions, and to interpret them as denoting properties of a system's "state", that is, as propositions about its location in configuration space. Because I have to consider non-deterministic systems from the outset, I often use the word "state" in a loose sense, to denote the position or configuration component of a contemplated state vector, whether or not it ever gets a deterministic completion. The set of logical features {a_1, ..., a_n} provides a basis for generating an n-dimensional "universe of discourse" that I denote as [a_1, ..., a_n]. It is useful to consider each universe of discourse as a unified categorical object that incorporates both the set of points <|a_1, ... , a_n|> and the set of propositions f : <|a_1, ..., a_n|> -> B that are implicit with the ordinary picture of a venn diagram on n features. Thus, we may regard the universe of discourse [a_1, ..., a_n] as an ordered pair having the type (B^n, (B^n - >B)), and we may abbreviate this last type designation as (B^n +-> B), or even more succinctly as [B^n]. (NB. I am using "<| ... |>" as "generator brackets".) Table 2 exhibits the scheme of notation I use to formalize the domain of propositional calculus, corresponding to the logical content of truth tables and venn diagrams. Although it overworks the square brackets a bit, I also use either one of the equivalent notations [n] or #n# to denote the data type of a finite set on n elements. Table 2. Fundamental Notations for Propositional Calculus o---------o-------------------o-------------------o-------------------o | Symbol | Notation | Description | Type | o---------o-------------------o-------------------o-------------------o | !A! | {a_1, ..., a_n} | Alphabet | [n] = #n# | o---------o-------------------o-------------------o-------------------o | A_i | {(a_i), a_i} | Dimension i | B | o---------o-------------------o-------------------o-------------------o | A | <|!A!|> | Set of cells, | B^n | | | <|a_i, ..., a_n|> | coordinate tuples,| | | | {<a_i, ..., a_n>} | interpretations, | | | | A_1 x ... x A_n | points, or vectors| | | | Prod_i A_i | in the universe | | o---------o-------------------o-------------------o-------------------o | A* | (hom : A -> B) | Linear functions | (B^n)* = B^n | o---------o-------------------o-------------------o-------------------o | A^ | (A -> B) | Boolean functions | B^n -> B | o---------o-------------------o-------------------o-------------------o | A% | [!A!] | Universe of Disc. | (B^n, (B^n -> B)) | | | (A, A^) | based on features | (B^n +-> B) | | | (A +-> B) | {a_1, ..., a_n} | [B^n] | | | (A, (A -> B)) | | | | | [a_1, ..., a_n] | | | o---------o-------------------o-------------------o-------------------o o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D3 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Qualitative Logic and Quantitative Analogy | Logical, however, is used in a third sense, which is at once more | vital and more practical; to denote, namely, the systematic care, | negative and positive, taken to safeguard reflection so that it | may yield the best results under the given conditions. | | John Dewey, 'How We Think', [Dew, 56] These concepts and notations can now be explained in greater detail. In order to begin as simply as possible, I distinguish two levels of analysis and set out initially on the easier path. On the first level of analysis, I take spaces like B, B^n, and (B^n->B) at face value and treat them as the primary objects of interest. On the second level of analysis, I use these spaces as coordinate charts for talking about points and functions in more fundamental spaces. A pair of spaces, of types B^n and (B^n->B), give typical expression to everything that we commonly associate with the ordinary picture of a venn diagram. The dimension, n, counts the number of "circles" or simple closed curves that are inscribed in the universe of discourse, corresponding to its relevant logical features or basic propositions. Elements of type B^n correspond to what are often called propositional "interpretations" in logic, that is, the different assignments of truth values to sentence letters. Relative to a given universe of discourse, these interpretations are visualized as its "cells", in other words, the smallest enclosed areas or undivided regions of the venn diagram. The functions f : B^n -> B correspond to the different ways of shading the venn diagram to indicate arbitrary propositions, regions, or sets. Regions included under a shading indicate the "models", and regions excluded represent the "non-models" of a proposition. To recognize and formalize the natural cohesion of these two layers of concepts into a single universe of discourse, I introduce the type notations [B^n] = B^n +-> B to stand for the pair of types (B^n, (B^n -> B)). The resulting "stereotype" serves to frame the universe of discourse as a unified categorical object, and makes it subject to prescribed sets of evaluations and transformations (categorical morphisms or "arrows") that affect the universe of discourse as an integrated whole. Most of the time we can serve the algebraic, geometric, and logical interests of our study without worrying about their occasional conflicts and incidental divergences. The conventions and definitions already set down will continue to cover most of the algebraic and functional aspects of our discussion, but to handle the logical and qualitative aspects we will need to add a few more. In general, abstract sets may be denoted by gothic, greek, or script capital variants of A, B, C, and so on, with elements denoted by a corresponding set of subscripted letters in plain lower case, for example, !A! = {a_i}. Most of the time, a set such as !A! = {a_i} will be employed as the "alphabet" of a formal language. These alphabet letters serve to name the logical features (properties or proposition) that generate a particular universe of discourse. When we want to discuss the particular features of a universe of discourse, beyond the abstract designation of a type like (B^n +-> B), then we may use the following notations. If !A! = {a_1, ..., a_n} is an alphabet of logical features, then A = <|!A!|> = <|a_1, ..., a_n|> is the set of interpretations, A^ = (A -> B) is the set of propositions, and A% = [!A!] = [a_1, ..., a_n] is the combination of these interpretations and propositions into the universe of discourse that is based on the features {a_1, ..., a_n}. As always, especially in concrete examples, these rules may be dropped whenever necessary, reverting to a free assortment of feature labels. However, when we need to talk about the logical aspects of a space that is already named as a vector space, it will be necessary to make special provisions. At any rate, these elaborations can be deferred until actually needed. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D4 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Philosophy of Notation: Formal Terms and Flexible Types | Where number is irrelevant, regimented mathematical technique has hitherto | tended to be lacking. Thus it is that the progress of natural science has | depended so largely upon the discernment of measurable quantity of one sort | or another. | | W.V. Quine, 'Mathematical Logic', [Qui, 7] For much of our discussion propositions and boolean functions are treated as the same formal objects, or as different interpretations of the same formal calculus. This rule of has exceptions, though. There is a distinctively logical interest in the use of propositional calculus that is not exhausted by its functional interpretation. It is part of our task in this study to deal with these uniquely logical characteristics as they present themselves both in our subject matter and in our formal calculus. Just to provide a hint of what's at stake: In logic, as opposed to the more imaginative realms of mathematics, we consider it a good thing to always know what we are talking about. Where mathematics encourages tolerance for uninterpreted symbols as intermediate terms, logic exerts a keener effort to interpret directly each oblique carrier of meaning, no matter how slight, and to unfold the complicities of every indirection in the flow of information. Translated into functional terms, this means that we want to maintain a continual, immediate, and persistent sense of both the converse relation f^(-1) c B x B^n, or what is the same thing, f^(-1) : B -> Pow(B^n), and the "fibers" or inverse images f^(-1)(0) and f^(-1)(1), associated with each boolean function f : B^n -> B that we use. In practical terms, the desired implementation of a propositional interpreter should incorporate our intuitive recognition that the induced partition of the functional domain into level sets f^(-1)(b), for b in B, is part and parcel of understanding the denotative uses of each propositional function f. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D5 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Special Classes of Propositions It is important to remember that the coordinate propositions {a_i}, besides being projection maps a_i : B^n -> B, are propositions on an equal footing with all others, even though employed as a basis in a particular moment. This set of n propositions may sometimes be referred to as the "basic" or "simple" propositions that found the universe of discourse. As typical and collective notations, we may use the forms {a_i : B^n -> B} = (B^n -i-> B) = (B^n -:> B) to indicate the adoption of a set of a_i as a basis for discourse. Among the 2^2^n propositions or functions in (B^n -> B) are several fundamental sets of 2^n propositions each that take on special forms with respect to a given basis !A! = {a_i}. Three of these forms are especially common, the "linear", the "positive", and the "singular" propositions. Each set is naturally parameterized by the coordinate vectors in B^n and falls into n+1 ranks, with a binomial coefficient C(n, k) giving the number of propositions that have rank or weight k. The "linear" propositions, {hom : B^n -> B} = (B^n -h-> B) = (B^n ++> B), may be expressed as sums of the following form: Sum_i e_i = e_1 + ... + e_n where e_i = a_i or e_i = 0. The "positive" propositions, {pos : B^n -> B} = (B^n -p-> B) = (B^n >=> B), may be expressed as products of the following form: Prod_i e_i = e_1 * ... * e_n where e_i = a_i or e_i = 1. The "singular" propositions, {x : B^n -> B} = (B^n -s-> B) = (B^n ::> B), may be expressed as products of the following form: Prod_i e_i = e_1 * ... * e_n where e_i = a_i or e_i = (a_i). In each case the rank k ranges from 0 to n and counts the number of positive appearances of coordinate propositions a_i in the resulting expression. For example, for n = 3, the linear proposition of rank 0 is "0", the positive proposition of rank 0 is "1", and the singular proposition of rank 0 is "(a_1)(a_2)(a_3)". The coordinate projections or simple propositions a_i : B^n -> B are both linear and positive. So these two kinds of propositions, the linear or the positive, may be viewed as two different ways of generalizing the class of simple projections. The linear and the positive propositions are generated by taking boolean sums and products, respectively, over selected subsets of the basic propositions in {a_i}. Therefore, each set of functions can be parameterized by the subsets J of the basic index set I = {1, ..., n}. Let us define A_J as the subset of A that is given by {a_i : i in J}. Then we may comprehend the action of the linear and the positive propositions in the following terms: 1. The linear proposition hom_J : B^n -> B evaluates each cell x of B^n by looking at x's coefficients with respect to the features that hom_J "likes", namely those in A_J, and then adds them up in B. Thus, hom_J (x) computes the parity of the number of features that x has in A_J, yielding one for odd and zero for even. Expressed in this idiom, hom_J (x) = 1 says that x seems "odd" (or "oddly true") to A_J, whereas hom_J (x) = 0 says that x seems "even" (or "evenly true") to A_J, so long as we recall that "zero times" is evenly often, too. 2. The positive proposition pos_J : B^n -> B evaluates each cell x of B^n by looking at x's coefficients with regard to the features that pos_J "likes", namely those in A_J, and then takes their product in B. Thus, pos_J (x) assesses the unanimity of the multitude of features that x has in A_J, yielding one for all and aught for else. In these consensual or contractual terms, pos_J (x) = 1 means that x is "AOK" or congruent with all of the conditions of A_J, while pos_J (x) = 0 means that x defaults or dissents from some condition of A_J. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D6 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Basis Relativity and Type Ambiguity Finally, two things are important to keep in mind with regard to the simplicity, linearity, positivity, and singularity of propositions. First, all of these properties are relative to a particular basis. For example, a singular proposition with respect to a basis !A! will not remain singular if A is extended by a number of new and independent features. Even if we stick to the original set of pairwise options {a_i} |_| {(a_i)} to select a new basis, the sets of linear and positive propositions are determined by the choice of simple propositions, and this determination is tantamount to the conventional choice of a cell as origin. Second, the singular propositions B ::> B, picking out as they do a single cell or a coordinate tuple of B^n, become the carriers or the vehicles of a certain type-ambiguity that vacillates between the dual forms B^n and (B^n ::> B) and infects the whole hierarchy of types built on them. In plainer language, the terms that signify the interpretations x : B^n and the singular propositions x : B^n ::> B are fully equivalent in information, and this means that every token of the type B^n can be reinterpreted as an appearance of the subtype B^n ::> B. And vice versa, the two types can be exchanged with each other everywhere that they turn up. In practical terms, this allows the use of singular propositions as a way of denoting points, forming an alternative to coordinate tuples. For example, relative to the universe of discourse [a_1, a_2, a_3] the singular proposition a_1 a_2 a_3 : B^3 ::> B could be explicitly retyped as a_1 a_2 a_3 : B^3 to indicate the point <1, 1, 1>, but in most cases the proper interpretation could be gathered from context. Both notations remain dependent on a particular basis, but the code that is generated under the singular option has the advantage in its self-commenting features, in other words, it constantly reminds us of its basis in the process of denoting points. When the time comes to put a multiplicity of different bases into play, and to search for objects and properties that remain invariant under the transformations between them, this infinitesimal potential advantage may well evolve into an overwhelming practical necessity. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D7 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o The Analogy Between Real and Boolean Types | Measurement consists in correlating our subject matter with the series of | real numbers; and such correlations are desirable because, once they are | set up, all the well-worked theory of numerical mathematics lies ready at | hand as a tool for our further reasoning. | | W.V. Quine, 'Mathematical Logic', [Qui, 7] There are two further reasons why I am spending so much time on a careful treatment of types, and they both have to do with our being able to take full computational advantage of certain dimensions of flexibility in the types that apply to terms. First, the domains of differential geometry and logic programming are connected by analogies between real and boolean types of the same pattern. Second, the types involved in these patterns have important isomorphisms connecting them that apply on both the real and the boolean sides of the picture. Amazingly enough, these isomorphisms are themselves schematized by the axioms and theorems of propositional logic. This fact is known as the "propositions as types" analogy or the Curry-Howard isomorphism [How]. In another formulation it says that terms are to types as proofs are to propositions. This principle seems to have more implications for our subject than I can fully comprehend at present, though I sense that they must be crucial. (Cf. [LaS, 42-46] and [SeH] for a good discussion and further references.) To anticipate the bearing of these issues on our immediate topic, Table 3 sketches a partial overview of the Real to Boolean analogy that may serve to illustrate the paradigm that I have in mind. Table 3. Analogy of Real and Boolean Types o-------------------------o-------------------------o-------------------------o | Real Domain R | <-> | Boolean Domain B | o-------------------------o-------------------------o-------------------------o | R^n | Basic Space | B^n | o-------------------------o-------------------------o-------------------------o | R^n -> R | Function Space | B^n -> B | o-------------------------o-------------------------o-------------------------o | (R^n -> R) -> R | Tangent Vector | (B^n -> B) -> B | o-------------------------o-------------------------o-------------------------o | R^n -> ((R^n -> R) -> R)| Vector Field | B^n -> ((B^n -> B) -> B)| o-------------------------o-------------------------o-------------------------o | (R^n x (R^n -> R)) -> R | ditto | (B^n x (B^n -> B)) -> B | o-------------------------o-------------------------o-------------------------o | ((R^n -> R) x R^n) -> R | ditto | ((B^n -> B) x B^n) -> B | o-------------------------o-------------------------o-------------------------o | (R^n -> R) -> (R^n -> R)| Derivation | (B^n -> B) -> (B^n -> B)| o-------------------------o-------------------------o-------------------------o | R^n -> R^m | Basic Transformation | B^n -> B^m | o-------------------------o-------------------------o-------------------------o | (R^n -> R) -> (R^m -> R)| Function Transformation | (B^n -> B) -> (B^m -> B)| o-------------------------o-------------------------o-------------------------o | ... | ... | ... | o-------------------------o-------------------------o-------------------------o The Table exhibits a sample of likely parallels between the real and boolean domains. The central column gives a selection of terminology that I borrow from typical usage in differential geometry and extend in its meaning to the logical side of the Table. These are the varieties of spaces that come up when we turn to analyzing the dynamics of processes that followe their courses through the states of an arbitrary space X. Moreover, when it becomes necessary to approach situations of overwhelming dynamic complexity in a succession of qualitative reaches, then the methods of logic that are afforded by the boolean domains, with their declarative means of synthesis and deductive modes of analysis, supply a natural battery of tools for the task. It is usually expedient to take these spaces two at a time, in dual pairs of the form X and (X -> K). In general, one creates pairs of type schemas by replacing any space X with its dual (X -> K), for example, pairing the type X -> Y with the type (X -> K) -> (Y -> K), and X x Y with (X -> K) x (Y -> K). Here, I am using the word "dual" in its larger sense to mean all of the functionals, not just the linear ones. Given any function f : X -> K, the "converse" or inverse relation corresponding to f is denoted as f^(-1), and the subsets of X that are defined by f^(-1)(k), taken over k in K, are called the "fibers" or the "level sets" of the function f. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D8 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Theory of Control and Control of Theory | You will hardly know who I am or what I mean, | But I shall be good health to you nevertheless, | And filter and fibre your blood. | | Walt Whitman, 'Leaves of Grass', [Whi, 88] In the boolean context, a function f : X -> B is tantamount to a "proposition" about elements of X, and the elements of X constitute the "interpretations" of that proposition. The fiber f^(-1)(1) comprises the set of "models" of f, or examples of elements in X satisfying the proposition f. The fiber f^(-1)(0) collects the complementary set of "anti-models", or the exceptions to the proposition f that exist in X. Of course, the space of functions (X -> B) is isomorphic to the set of all subsets of X, called the "power set" of X and often denoted as Pow(X) or 2^X. The operation of replacing X by (X -> B) in a type schema corresponds to a certain shift of attitude towards the space X, in which one passes from a focus on the ostensibly individual elements of X to a concern with the states of information and uncertainty that one possesses about objects and situations in X. The conceptual obstacles in the path of this transition can be smoothed over by using singular functions (X ::> B) as stepping stones. First of all, it's an easy step from an element x of type B^n to the equivalent information of a singular proposition x : X ::> B, and then only a small jump of generalization remains to reach the type of an arbitrary proposition f : X -> B, perhaps understood to indicate a relaxed constraint on the singularity of points or a neighborhood circumscribing the original x. I have frequently discovered this to be a useful transformation, communicating between the "objective" and the "intentional" perspectives, in spite perhaps of the open objection that this distinction is transient in the mean time and ultimately superficial. It is hoped that this measure of flexibility, allowing us to stretch a point into a proposition, can be useful in the examination of inquiry driven systems, where the differences between empirical, intentional, and theoretical propositions constitute the discrepancies and the distributions that drive experimental activity. I can give this model of inquiry a cybernetic cast by realizing that theory change and theory evolution, as well as the choice and the evaluation of experiments, are actions that are taken by a system or its agent in response to the differences that are detected between observational contents and theoretical coverage. All of the above notwithstanding, there are several points that distinguish these two tasks, namely, the "theory of control" and the "control of theory", features that are often obscured by too much precipitation in the quickness with which we understand their similarities. In the control of uncertainty through inquiry, some of the actuators that we need to be concerned with are axiom changers and theory modifiers, operators with the power to compile and to revise the theories that generate expectations and predictions, effectors that form and edit our grammars for the languages of observational data, and agencies that rework the proposed model to fit the actual sequences of events and the realized relationships of values that are observed in the environment. Moreover, when steps must be taken to carry out an experimental action, there must be something about the particular shape of our uncertainty that guides us in choosing what directions to explore, and this impression is more than likely influenced by previous accumulations of experience. Thus it must be anticipated that much of what goes into scientific progress, or any sustainable effort toward a goal of knowledge, is necessarily predicated on long term observation and modal expectations, not only on the more local or short term prediction and correction. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D9 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Propositions as Types and Higher Order Types The arrangement of types collected in Table 3 can serve as a good introduction to several ideas about "higher order propositional expressions" (HOPE's) and also about the "propositions as types" (PAT) isomorphism. Table 3. Analogy of Real and Boolean Types o-------------------------o-------------------------o-------------------------o | Real Domain R | <-> | Boolean Domain B | o-------------------------o-------------------------o-------------------------o | R^n | Basic Space | B^n | o-------------------------o-------------------------o-------------------------o | R^n -> R | Function Space | B^n -> B | o-------------------------o-------------------------o-------------------------o | (R^n -> R) -> R | Tangent Vector | (B^n -> B) -> B | o-------------------------o-------------------------o-------------------------o | R^n -> ((R^n -> R) -> R)| Vector Field | B^n -> ((B^n -> B) -> B)| o-------------------------o-------------------------o-------------------------o | (R^n x (R^n -> R)) -> R | ditto | (B^n x (B^n -> B)) -> B | o-------------------------o-------------------------o-------------------------o | ((R^n -> R) x R^n) -> R | ditto | ((B^n -> B) x B^n) -> B | o-------------------------o-------------------------o-------------------------o | (R^n -> R) -> (R^n -> R)| Derivation | (B^n -> B) -> (B^n -> B)| o-------------------------o-------------------------o-------------------------o | R^n -> R^m | Basic Transformation | B^n -> B^m | o-------------------------o-------------------------o-------------------------o | (R^n -> R) -> (R^m -> R)| Function Transformation | (B^n -> B) -> (B^m -> B)| o-------------------------o-------------------------o-------------------------o | ... | ... | ... | o-------------------------o-------------------------o-------------------------o First, observe that the type of a "Tangent Vector at a Point", also known as a "directional derivative" at that point, has the form (K^n -> K) -> K, where K is the chosen ground field, in the present case either R or B. At a point in a space of type K^n, a directional derivative operator !q! takes a function on that space, an f of type (K^n -> K), and maps it to a ground field value of type K. This value is known as the "derivative" of f in the direction !q! [Che46, 76-77]. In the boolean case, !q! : (B^n -> B) -> B has the form of a proposition about propositions, in other words, a proposition of the next higher type. Next, by way of illustrating the propositions as types theme, consider a proposition of the form X => (Y => Z). One knows from propositional calculus that this is logically equivalent to a proposition of the form (X & Y) => Z. But this equivalence should remind us of the functional isomorphism that exists between a construction of the type X -> (Y -> Z) and a construction of the type (X x Y) -> Z. The propositions as types analogy permits us to take a functional type like this and, under the right conditions, replace the functional arrows "->" and products "x" with the respective logical arrows "=>" and products "&". Accordingly, viewing the result as a proposition, we can employ axioms and theorems of propositional calculus to suggest appropriate isomorphisms among the categorical and functional constructions. Finally, examine the middle four rows of Table 3. These display a series of isomorphic types that stretch from the categories that are labeled "Vector Field" to those that are labeled "Derivation". A "vector field", also known as an "infinitesimal transformation", associates a tangent vector at a point with each point of a space. In symbols, a vector field is a function !X! : X -> |_| !X!_x that assigns to each point x of the space X a tangent vector !X!_x to X at that point [Che46, 82-83]. If X is of type K^n, then !X! is of type K^n -> ((K^n -> K) -> K). This has the pattern X -> (Y -> Z), with X = K^n, Y = (K^n -> K), and Z = K. Applying the propositions as types analogy, one can follow this pattern through a series of metamorphoses from the type of a vector field to the type of a derivation, as traced out in Table 4. Observe how the function f : X -> K, associated with the place of Y in the pattern, moves through its paces from the second to the first position. In this way, the vector field !X!, initially viewed as attaching each tangent vector !X!_x to the site x where it acts in X, now comes to be seen as acting on each scalar potential f : X -> K like a generalized species of differentiation, producing another function !X!f : X -> K of the same type. Table 4. An Equivalence Based on the Propositions as Types Analogy o-------------------------o------------------------o--------------------------o | Pattern | Construction | Instance | o-------------------------o------------------------o--------------------------o | X -> (Y -> Z) | Vector Field | K^n -> ((K^n -> K) -> K) | o-------------------------o------------------------o--------------------------o | (X x Y) -> Z | | (K^n x (K^n -> K)) -> K | o-------------------------o------------------------o--------------------------o | (Y x X) -> Z | | ((K^n -> K) x K^n) -> K | o-------------------------o------------------------o--------------------------o | Y -> (X -> Z) | Derivation | (K^n -> K) -> (K^n -> K) | o-------------------------o------------------------o--------------------------o o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D10 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Reality at the Threshold of Logic | But no science can rest entirely on measurement, and many | scientific investigations are quite out of reach of that | device. To the scientist longing for non-quantitative | techniques, then, mathematical logic brings hope. | | W.V. Quine, 'Mathematical Logic', [Qui, 7] Table 5 accumulates an array of notation that I hope will not be too distracting. Some of it is rarely needed, but has been filled in for the sake of completeness. Its purpose is simple, to give literal expression to the visual intuitions that come with venn diagrams, and to help build a bridge between our qualitative and quantitative outlooks on dynamic systems. NB. I'm trying to keep the Asciification of the text as simple as possible this time around, so I will use emphasis bars !...! around characters in a context-dependent way, sometimes for Gothic and sometimes for Greek. Thus, I will rely on the reader's own recognizance to discriminate between entities like a "Script X" alphabet !X! = {x_1, ..., x_n} and a "Greek Chi" vector field !X!. Also, the original version of the Table below used underlined variants of the characters in the middle column, to suggest quantities rising above the relevant threshold. In this copy, I use grave markers `...` around the thresheld symbols instead of underscoring them. Finally, there does not seem to be any way to avoid the clash of symbols between the stars, that is, the '*' that is used in algebra to denote the dual space of linear functionals, and the '*' that is used in formal language theory to denote the set of all finite sequences over an alphabet. Table 5. A Bridge Over Troubled Waters o-------------------------o-------------------------o-------------------------o | Linear Space | Liminal Space | Logical Space | o-------------------------o-------------------------o-------------------------o | | | | | !X! | !`X`! | !A! | | | | | | {x_1, ..., x_n} | {`x`_1, ..., `x`_n} | {a_1, ..., a_n} | | | | | | cardinality n | cardinality n | cardinality n | o-------------------------o-------------------------o-------------------------o | | | | | X_i | `X`_i | A_i | | | | | | <|x_i|> | {(`x`_i), `x`_i} | {(a_i), a_i} | | | | | | isomorphic to K | isomorphic to B | isomorphic to B | o-------------------------o-------------------------o-------------------------o | | | | | X | `X` | A | | | | | | <|!X!|> | <|!`X`!|> | <|!A!|> | | | | | | <|x_1, ..., x_n|> | <|`x`_1, ..., `x`_n|> | <|a_1, ..., a_n|> | | | | | | {<x_1, ..., x_n>} | {<`x`_1, ..., `x`_n>} | {<a_1, ..., a_n>} | | | | | | X_1 x ... x X_n | `X`_1 x ... x `X`_n | A_1 x ... x A_n | | | | | | Prod_i X_i | Prod_i `X`_i | Prod_i A_i | | | | | | isomorphic to K^n | isomorphic to B^n | isomorphic to B^n | o-------------------------o-------------------------o-------------------------o | | | | | X* | `X`* | A* | | | | | | (hom : X -> K) | (hom : `X` -> B) | (hom : A -> B) | | | | | | isomorphic to K^n | isomorphic to B^n | isomorphic to B^n | o-------------------------o-------------------------o-------------------------o | | | | | X^ | `X`^ | A^ | | | | | | (X -> K) | (`X` -> B) | (A -> B) | | | | | | isomorphic to (K^n -> K)| isomorphic to (B^n -> B)| isomorphic to (B^n -> B)| o-------------------------o-------------------------o-------------------------o | | | | | X% | `X`% | A% | | | | | | [!X!] | [!`X`!] | [!A!] | | | | | | [x_1, ..., x_n] | [`x`_1, ..., `x`_n] | [a_1, ..., a_n] | | | | | | (X, X^) | (`X`, `X`^) | (A, A^) | | | | | | (X +-> K) | (`X` +-> B) | (A +-> B) | | | | | | (X, (X -> K)) | (`X`, (`X` -> B)) | (A, (A -> B)) | | | | | | isomorphic to: | isomorphic to: | isomorphic to: | | | | | | (K^n, (K^n -> K) | (B^n, (B^n -> B) | (B^n, (B^n -> K) | | | | | | (K^n +-> K) | (B^n +-> B) | (B^n +-> B) | | | | | | [K^n] | [B^n] | [B^n] | o-------------------------o-------------------------o-------------------------o The left side of the Table collects mostly standard notation for an n-dimensional vector space over a field K. The right side of the table repeats the first elements of a notation that I sketched above, to be used in further developments of propositional calculus. (I plan to use this notation in the logical analysis of neural network systems.) The middle column of the table is designed as a transitional step from the case of an arbitrary field K, with a special interest in the continuous line R, to the qualitative and discrete situations that are instanced and typified by B. I now proceed to explain these concepts in more detail. The two most important ideas developed in the table are: 1. The idea of a universe of discourse, which includes both a space of "points" and a space of "maps" on those points. 2. The idea of passing from a more complex universe to a simpler universe by a process of "thresholding" each dimension of variation down to a single bit of information. For the sake of concreteness, let us suppose that we start with a continuous n-dimensional vector space like X = <|x_1, ..., x_n|> ~=~ R^n. The coordinate system !X! = {x_i} is a set of maps x_i : R^n -> R, also known as the coordinate projections. Given a "dataset" of points x in R^n, there are numerous ways of sensibly reducing the data down to one bit for each dimension. One strategy that is general enough for our present purposes is as follows. For each i we choose an n-ary relation L_i on R, that is, a subset of R^n, and then we define the i^th threshold map, or "limen" `x`_i as follows: `x`_i : R^n -> B such that: `x`_i (x) = 1 if x in L_i, `x`_i (x) = 0 if otherwise. In other notations that are sometimes used, the operator <chi>( ) or the corner brackets |^...^| can be used to denote a "characteristic function", that is, a mapping from statements to their truth values, given as elements of B. Finally, it is not uncommon to use the name of the relation itself as a predicate that maps n-tuples into truth values. In each of these notations, the above definition could be expressed as follows: `x`_i (x) = <chi>(x in L_i) = |^ x in L_i ^| = L_i (x). Notice that, as defined here, there need be no actual relation between the n-dimensional subsets {L_i} and the coordinate axes corresponding to {x_i}, aside from the circumstance that the two sets have the same cardinality. In concrete cases, though, one usually has some reason for associating these "volumes" with these "lines", for instance, L_i is bounded by some hyperplane that intersects the i^th axis at a unique threshold value r_i in R. Often, the hyperplane is chosen normal to the axis. In recognition of this motive, let us make the following convention. When the set L_i has points on the i^th axis, that is, points of the form <0, ..., 0, r_i, 0, ..., 0> where only the x_i coordinate is possibly non-zero, we may pick any one of these coordinate values as a parametric index of the relation. In this case we say that the indexing is "real", otherwise the indexing is "imaginary". For a knowledge based system X, this should serve once again to mark the distinction between "acquaintance" and "opinion". States of knowledge about the location of a system or about the distribution of a population of systems in a state space X = R^n can now be expressed by taking the set !`X`! = {`x`_i} as a basis of logical features. In picturesque terms, one may think of the underscore [here, the grave accents] and the subscript as combining to form a subtextual spelling for the i^th threshold map. This can help to remind us that the "threshold operator" `( )`_i acts on x by setting up a kind of a "hurdle" for it. In this interpretation, the coordinate proposition `x`_i asserts that the representative point x resides "above" the i^th threshold. Primitive assertions of the form `x`_i (x) can then be negated and joined by means of propositional connectives in the usual ways to provide information about the state x of a contemplated system or a statistical ensemble of systems. Parentheses "( )" may be used to indicate negation. Eventually one discovers the usefulness of the k-ary "just one false" operators of the form "( , , , )", as treated in earlier reports. This much tackle generates a space of points (cells, interpretations), `X` = <|!`X`!|> ~=~ B^n, and a space of functions (regions, propositions), `X`^ ~=~ (B^n -> B). Together these form a new universe of discourse `X`% = [!`X`!] of the type (B^n, (B^n -> B)), which we may abbreviate as B^n +-> B, or most succinctly as [B^n]. The square brackets have been chosen to recall the rectangular frame of a venn diagram. In thinking about a universe of discourse it is a good idea to keep this picture in mind, where we constantly think of the elementary cells `x`, the defining features `x`_i, and the potential shadings f : `X` -> B, all at the same time, remaining aware of the arbitrariness of the way that we choose to inscribe our distinctions in the medium of a continuous space. Finally, let X* denote the space of linear functions, (hom : X -> K), which in the finite case has the same dimensionality as X, and let the same notation be extended across the table. We have just gone through a lot of work, apparently doing nothing more substantial than spinning a complex spell of notational devices through a labyrinth of baffled spaces and baffling maps. The reason for doing this was to bind together and to constitute the intuitive concept of a universe of discourse into a coherent categorical object, the kind of thing, once grasped, which can be turned over in the mind and considered in all its manifold changes and facets. The effort invested in these preliminary measures is intended to pay off later, when we need to consider the state transformations and the time evolution of neural network systems. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D11 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Tables of Propositional Forms | To the scientist longing for non-quantitative techniques, then, | mathematical logic brings hope. It provides explicit techniques | for manipulating the most basic ingredients of discourse. | | W.V. Quine, 'Mathematical Logic', [Qui, 7-8] To prepare for the next phase of discussion, Tables 6 and 7 collect and summarize all of the propositional forms on one and two variables. These propositional forms are represented over bases of boolean variables as complete sets of boolean-valued functions. Adjacent to their names and specifications are listed what are roughly the simplest expressions in the "cactus language", the particular syntax for propositional calculus that I use in formal and computational contexts. For the sake of orientation, the English paraphrases and the more common notations are listed in the last two columns. As simple and circumscribed as these low-dimensional universes may appear to be, a careful exploration of their differential extensions will involve us in complexities sufficient to demand our attention for some time to come. Propositional forms on one variable correspond to boolean functions f : B^1 -> B. In Table 6 these functions are listed in a variant form of truth table, one which rotates the axes of the usual arrangement. Each function f_i is indexed by the string of values that it takes on the points of the universe X% = [x] ~=~ B^1. The binary index generated in this way is converted to its decimal equivalent, and these are used as conventional names for the f_i, as shown in the first column of the Table. In their own right the 2^1 points of the universe X% are coordinated as a space of type B^1, this in light of the universe X% being a functional domain where the coordinate projection x takes on its values in B. Table 6. Propositional Forms on One Variable o---------o---------o---------o----------o------------------o----------o | L_1 | L_2 | L_3 | L_4 | L_5 | L_6 | | | | | | | | | Decimal | Binary | Vector | Cactus | English | Ordinary | o---------o---------o---------o----------o------------------o----------o | | x : 1 0 | | | | o---------o---------o---------o----------o------------------o----------o | | | | | | | | f_0 | f_00 | 0 0 | ( ) | false | 0 | | | | | | | | | f_1 | f_01 | 0 1 | (x) | not x | ~x | | | | | | | | | f_2 | f_10 | 1 0 | x | x | x | | | | | | | | | f_3 | f_11 | 1 1 | (( )) | true | 1 | | | | | | | | o---------o---------o---------o----------o------------------o----------o Propositional forms on two variables correspond to boolean functions f : B^2 -> B. In Table 7 each function f_i is indexed by the values that it takes on the points of the universe X% = [x, y] ~=~ B^2. Converting the binary index thus generated to a decimal equivalent, we obtain the functional nicknames that are listed in the first column. The 2^2 points of the universe X% are coordinated as a space of type B^2, as indicated under the heading of the Table, where the coordinate projections x and y run through the various combinations of their values in B. Table 7. Propositional Forms On Two Variables o---------o---------o---------o----------o------------------o----------o | L_1 | L_2 | L_3 | L_4 | L_5 | L_6 | | | | | | | | | Decimal | Binary | Vector | Cactus | English | Ordinary | o---------o---------o---------o----------o------------------o----------o | | x : 1 1 0 0 | | | | | | y : 1 0 1 0 | | | | o---------o---------o---------o----------o------------------o----------o | | | | | | | | f_0 | f_0000 | 0 0 0 0 | () | false | 0 | | | | | | | | | f_1 | f_0001 | 0 0 0 1 | (x)(y) | neither x nor y | ~x & ~y | | | | | | | | | f_2 | f_0010 | 0 0 1 0 | (x) y | y and not x | ~x & y | | | | | | | | | f_3 | f_0011 | 0 0 1 1 | (x) | not x | ~x | | | | | | | | | f_4 | f_0100 | 0 1 0 0 | x (y) | x and not y | x & ~y | | | | | | | | | f_5 | f_0101 | 0 1 0 1 | (y) | not y | ~y | | | | | | | | | f_6 | f_0110 | 0 1 1 0 | (x, y) | x not equal to y | x + y | | | | | | | | | f_7 | f_0111 | 0 1 1 1 | (x y) | not both x and y | ~x v ~y | | | | | | | | | f_8 | f_1000 | 1 0 0 0 | x y | x and y | x & y | | | | | | | | | f_9 | f_1001 | 1 0 0 1 | ((x, y)) | x equal to y | x = y | | | | | | | | | f_10 | f_1010 | 1 0 1 0 | y | y | y | | | | | | | | | f_11 | f_1011 | 1 0 1 1 | (x (y)) | not x without y | x => y | | | | | | | | | f_12 | f_1100 | 1 1 0 0 | x | x | x | | | | | | | | | f_13 | f_1101 | 1 1 0 1 | ((x) y) | not y without x | x <= y | | | | | | | | | f_14 | f_1110 | 1 1 1 0 | ((x)(y)) | x or y | x v y | | | | | | | | | f_15 | f_1111 | 1 1 1 1 | (()) | true | 1 | | | | | | | | o---------o---------o---------o----------o------------------o----------o o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D12 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | Fire over water: | The image of the condition before transition. | Thus the superior man is careful | In the differentiation of things, | So that each finds its place. | | 'I Ching', Hexagram 64, [Wil, 249] A Differential Extension of Propositional Calculus This much preparation is enough to begin introducing my subject, if I excuse myself from giving full arguments for my definitional choices until some later stage. I am trying to develop a "differential theory of qualitative equations" that parallels the application of differential geometry to dynamic systems. The idea of a tangent vector is key to this work and a major goal is to find the right logical analogues of tangent spaces, bundles, and functors. The strategy is taken of looking for the simplest versions of these constructions that can be discovered within the realm of propositional calculus, so long as they serve to fill out the general theme. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D13 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Differential Propositions: The Qualitative Analogue of Differential Equations In order to define the differential extension of a universe of discourse [!A!], the initial alphabet !A! must be extended to include a collection of symbols for "differential features", or basic "changes" that are capable of occurring in [!A!]. Intuitively, these symbols may be construed as denoting primitive features of change, qualitative attributes of motion, or propositions about how things or points in [!A!] may change or move with respect to the features that are noted in the initial alphabet. Hence, let us define the corresponding "differential alphabet" or "tangent alphabet" as d!A! = {da_1, ... , da_n}, in principle, just an arbitrary alphabet of symbols, disjoint from the initial alphaber !A! = {a_1, ..., a_n}, that is intended to be interpreted in the way just indicated. It only remains to be understood that the precise interpretation of the symbols in d!A! is often conceived to be changeable from point to point of the underlying space A. (For all we know, the state space A might well be the state space of a language interpreter, one that is concerned, among other things, with the idiomatic meanings of the dialect generated by !A! and d!A!.) In the above terms, a typical "tangent space of A at a point x", frequently denoted as T_x (A), can be characterized as having the generic construction dA = <|d!A!|> = <|da_1, ..., da_n|>. Strictly speaking, the name "cotangent space" is probably more correct for this construction, but the fact that we take up spaces and their duals in pairs to form our universes of discourse allows our language to be pliable here. Proceeding as we did before with the base space A, we can analyze the individual tangent space at a point of A as a product of distinct and independent factors: dA = Prod_i dA_i = dA_1 x ... x dA_n. Here, dA_i is an alphabet of two symbols, dA_i = {(da_i), da_i}, where (da_i) is a symbol with the logical value of "not da_i". Each component dA_i has the type B, under the correspondence {(da_i), da_i} ~=~ {0, 1}. However, clarity is often served by acknowledging this differential usage with a superficially distinct type D, whose intension may be indicated as follows: D = {(da_i), da_i} = {same, different} = {stay, change} = {stop, step}. Viewed within a coordinate representation, spaces of type B^n and D^n may appear to be identical sets of binary vectors, but taking a view at this level of abstraction would be like ignoring the qualitative units and the diverse dimensions that distinguish position and momentum, or the different roles of quantity and impulse. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D14 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o An Interlude on the Path | There would have been no beginnings: | instead, speech would proceed from me, | while I stood in its path - a slender gap - | the point of its possible disappearance. | | Michel Foucault, 'The Discourse on Language', [Fou, 215] It may help to get a sense of the relation between B and D by considering the "path classifier" (or equivalence class of curves) approach to tangent vectors. As if by reflex, the thought of physical motion makes us cross over to a universe marked by the nominal character [!X!]. Given the boolean value system, a path in the space X = <|!X!|> is a map q : B -> X. In this case, the set of paths (B -> X) is isomorphic to the cartesian square X^2 = X x X, or the set of ordered pairs from X. We may analyze X^2 = {<u, v> : u, v in X} into two parts, specifically, the pairs that lie on and off the diagonal: X^2 = {<u, v> : u = v} |_| {<u, v> : u =/= v}. In symbolic terms, this partition may be expressed as: X^2 ~=~ Diag(X) + 2 * Comb(X, 2), where: Diag(X) = {<x, x> : x in X}, and where: Comb(X, k) = "X choose k" = {k-sets from X}, so that: Comb(X, 2) = {{u, v} : u, v in X}. We can now use the features in d!X! = {dx_1, ... , dx_n} to classify the paths of (B -> X) by way of the pairs in X^2. If X ~=~ B^n then a path in X has the form q : (B -> B^n) ~=~ B^n x B^n ~=~ B^2n ~=~ (B^2)^n. Intuitively, we want to map this (B^2)^n onto D^n by mapping each component B^2 onto a copy of D. But in our current situation "D" is just a name we give, or an accidental quality we attribute, to coefficient values in B when they are attached to features in d!X!. Therefore, define dx_i : X^2 -> B such that: dx_i (<u, v>) = (| x_i (u) , x_i (v) |) = x_i (u) + x_i (v) = x_i (v) - x_i (u). NB. In the above transcription, "(| ... , ... |)" is a "cactus lobe", signifying "just one false", in this case among two boolean variables, while "+" is boolean addition in the proper sense of addition in GF(2), and thus equivalent to "-", in the sense of adding the additive inverse. The above definition is equivalent to defining dx_i : (B -> X) -> B such that: dx_i (q) = (| x_i (q_0) , x_i (q_1) |) = x_i (q_0) + x_i (q_1) = x_i (q_1) - x_i (q_0), where q_b = q(b), for each b in B. Thus, the proposition dx_i is true of the path q = <u, v> exactly if the terms of q, the endpoints u and v, lie on different sides of the question x_i. Now we can use the language of features in <|d!X!|>, indeed the whole calculus of propositions in [d!X!], to classify paths and sets of paths. In other words, the paths can be taken as models of the propositions g : dX -> B. For example, the paths corresponding to Diag(X) fall under the description (dx_1)...(dx_n), which says that nothing changes among the set of features {x_1, ..., x_n}. Finally, a few words of explanation may be in order. If this concept of a path appears to be described in a roundabout fashion, it is because I am trying to avoid using any assumption of vector space properties for the space X which contains its range. In many ways the treatment is still unsatisfactory, but improvements will have to wait for the introduction of substitution operators acting on singular propositions. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D15 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o The Extended Universe of Discourse | At the moment of speaking, I would like to have perceived a nameless voice, | long preceding me, leaving me merely to enmesh myself in it, taking up its | cadence, and to lodge myself, when no one was looking, in its interstices | as if it had paused an instant, in suspense, to beckon to me. | | Michel Foucault, 'The Discourse on Language', [Fou, 215] Next, we define the so-called "extended alphabet" or "bundled alphabet" E!A! as: E!A! = !A! |_| d!A! = {a_1, ..., a_n, da_1, ..., da_n}. This supplies enough material to construct the "differential extension" EA, or the "tangent bundle" over the initial space A, in the following fashion: EA = A x dA = <|E!A!|> = <|!A! |_| d!A!|> = <|a_1, ..., a_n, da_1, ..., da_n|>, thus giving EA the type B^n x D^n. Finally, the tangent universe EA% = [E!A!] is constituted from the totality of points and maps, or interpretations and propositions, which are based on the extended set of features E!A!: EA% = [E!A!] = [a_1, ..., a_n, da_1, ..., da_n], thus giving the tangent universe E!A! the type (B^n x D^n +-> B) = (B^n x D^n, (B^n x D^n -> B). A proposition in the tangent universe [E!A!] is called a "differential proposition" and forms the analogue of a system of differential equations, constraints, or relations in ordinary calculus. With these constructions, to be specific, the differential extension EA and the differential proposition h : EA -> B, we have arrived, in concept at least, at one of the major subgoals of this study. At this juncture, I pause by way of summary to set another Table with the current crop of mathematical produce (Table 8). Table 8. Notation for the Differential Extension of Propositional Calculus o---------o-------------------o-------------------o-------------------o | Symbol | Notation | Description | Type | o---------o-------------------o-------------------o-------------------o | d!A! | {da_1, ..., da_n} | Alphabet of | [n] = #n# | | | | differential | | | | | features | | o---------o-------------------o-------------------o-------------------o | dA_i | {(da_i), da_i} | Differential | D | | | | dimension i | | o---------o-------------------o-------------------o-------------------o | dA | <|d!A!|> | Tangent space | D^n | | | <|da_i,...,da_n|> | at a point: | | | | {<da_i,...,da_n>} | Set of changes, | | | | dA_1 x ... x dA_n | motions, steps, | | | | Prod_i dA_i | tangent vectors | | | | | at a point | | o---------o-------------------o-------------------o-------------------o | dA* | (hom : dA -> B) | Linear functions | (D^n)* ~=~ D^n | | | | on dA | | o---------o-------------------o-------------------o-------------------o | dA^ | (dA -> B) | Boolean functions | D^n -> B | | | | on dA | | o---------o-------------------o-------------------o-------------------o | dA% | [d!A!] | Tangent universe | (D^n, (D^n -> B)) | | | (dA, dA^) | at a point of A%, | (D^n +-> B) | | | (dA +-> B) | based on the | [D^n] | | | (dA, (dA -> B)) | tangent features | | | | [da_1, ..., da_n] | {da_1, ..., da_n} | | o---------o-------------------o-------------------o-------------------o The adjectives "differential" or "tangent" are systematically attached to every construct based on the differential alphabet d!A!, taken by itself. Strictly speaking, we probably ought to call d!A! the set of "cotangent" features derived from !A!, but the only time this distinction really seems to matter is when we need to distinguish the tangent vectors as maps of type (B^n -> B) -> B from cotangent vectors as elements of type D^n. In like fashion, having defined E!A! = !A! |_| d!A!, we can systematically attach the adjective "extended" or the substantive "bundle" to all the constructs associated with this full complement of 2n features. Eventually we may want to extend our basic alphabet even further, to allow for discussion of higher order differential expressions. For those who want to run ahead, and would like to play through, I submit the following gamut of notation (Table 9). Table 9. Higher Order Differential Features o----------------------------------------o----------------------------------------o | | | | !A! = d^0.!A! = {a_1, ..., a_n} | E^0.!A! = d^0.!A! | | | | | d!A! = d^1.!A! = {da_1, ..., da_n} | E^1.!A! = d^0.!A! |_| d^1.!A! | | | | | d^k.!A! = {d^k.a_1,...,d^k.a_n}| E^k.!A! = d^0.!A! |_| ... |_| d^k.!A! | | | | | d*!A! = {d^0.!A!, ..., d^k.!A!, ...} | E^oo.!A! = d*!A! | | | | o----------------------------------------o----------------------------------------o o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D16 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Intentional Propositions | Do you guess I have some intricate purpose? | Well I have . . . . for the April rain has, and the mica on | the side of a rock has. | | Walt Whitman, 'Leaves of Grass', [Whi, 45] In order to analyze the behavior of a system at successive moments in time, while staying within the limitations of propositional logic, it is necessary to create independent alphabets of logical features for each moment of time that we contemplate using in our discussion. These moments have reference to typical instances and relative intervals, not actual or absolute times. For example, to discuss "velocities" (first order rates of change) we need to consider points of time in pairs. There are a number of natural ways of doing this. Given an initial alphabet, we could use its symbols as a lexical basis to generate successive alphabets of compound symbols, say, with temporal markers appended as suffixes. As a standard way of dealing with these situations, I produce the following scheme of notation, which extends any alphabet of logical features through as many temporal moments as a particular order of analysis may demand. The lexical operators p^k and Q^k are convenient in many contexts where the accumulation of prime symbols and union symbols would otherwise be cumbersome. Table 10. A Realm of Intentional Features o---------------------------------------o----------------------------------------o | | | | p^0.!A! = !A! = {a_1, ..., a_n} | Q^0.!A! = !A! | | | | | p^1.!A! = !A!' = {a_1', ..., a_n'} | Q^1.!A! = !A! |_| !A!' | | | | | p^2.!A! = !A!" = {a_1", ..., a_n"} | Q^2.!A! = !A! |_| !A!' |_| !A!" | | | | | ... ... ... | ... ... | | | | | p^k.!A! = {p^k.a_1, ..., p^k.a_n} | Q^k.!A! = !A! |_| ... |_| p^k.!A! | | | | o---------------------------------------o----------------------------------------o The resulting augmentations of our logical basis found a series of discursive universes that may be called the "intentional extension" of propositional calculus. The pattern of this extension is analogous to that of the differential extension, which was developed in terms of the operators d^k and E^k, and there is an obvious and natural relation between these two extensions that falls within our purview to explore. In contexts displaying this regular pattern, where a series of domains stretches up from an anchoring domain X through an indefinite number of higher reaches, I refer to a particular collection of domains based on X as "a realm of X", and when the succession exhibits a temporal aspect, "a reign of X". For the purposes of this discussion, let us define an "intentional proposition" as a proposition in the universe of discourse QX% = [Q!X!], in other words, a map q : QX -> B. The sense of this definition may be seen if we consider the following facts. First, the equivalence QX = X x X' motivates the following chain of isomorphisms between spaces: (QX -> B) ~=~ (X x X' -> B) ~=~ (X -> (X' -> B)) ~=~ (X' -> (X -> B)). Viewed in this light, an intentional proposition q may be rephrased as a map q : X x X' -> B, which judges the juxtaposition of states in X from one moment to the next. Alternatively, q may be parsed in two stages in two different ways, as q : X -> (X' -> B) and q : X' -> (X -> B), which associate to each point of X or X' a proposition about states in X' or X, respectively. In this way, an intentional proposition embodies a type of value system, in effect, a proposal that places a value on a collection of ends-in-view, or a project that evaluates a set of goals as regarded from each point of view in the state space of a system. In sum, the intentional proposition q indicates a method for the systematic selection of local goals. As a general form of description, we may refer to a map of the type q : Q^i.X -> B as an "i^th order intentional proposition". Naturally, when we speak of intentional propositions without qualification, we usually mean first order intentions. Many different realms of discourse have the same structure as the extensions that have been indicated here. From a strictly logical point of view, each new layer of terms is composed of independent logical variables that are no different in kind from those that go before, and each further course of logical atoms is treated like so many individual, but otherwise indifferent bricks by the prototype computer program that I use as a propositional interpreter. Thus, the names that I use to single out the differential and the intentional extensions, and the lexical paradigms that I follow to construct them, are meant to suggest the interpretations that I have in mind, but they can only hint at the extra meanings that human communicators may pack into their terms and inflections. As applied here, the word "intentional" is drawn from common use and may have little bearing on its technical use in other, more properly philosophical, contexts. I am merely using the complex of intentional concepts -- aims, ends, goals, objectives, purposes, and so on -- metaphorically to flesh out and vividly to represent any situation where one needs to contemplate a system in multiple aspects of state and destination, that is, its being in certain states and at the same time acting as if headed through certain states. If confusion arises, more neutral words like conative, contingent, discretionary, experimental, kinetic, progressive, tentative, or trial would probably serve as well. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D17 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Life on Easy Street | Failing to fetch me at first keep encouraged, | Missing me one place search another, | I stop some where waiting for you | | Walt Whitman, 'Leaves of Grass', [Whi, 88] The finite character of the extended universe [E!A!] makes the problem of solving differential propositions relatively straightforward, at least, in principle. The solution set of the differential proposition q : EA -> B is the set of models (q^(-1))(1) in EA. Finding all of the models of q, the extended interpretations in EA that satisfy q, can be carried out by a finite search. Being in possession of complete algorithms for propositional calculus theorem proving makes the analytic task fairly simple in principle, though the question of efficiency in the face of arbitrary complexity may always remain another matter entirely. While the fact that propositional satisfiability is NP-complete may be discouraging for the prospects of a single efficient algorithm that covers the whole space of [E!A!] with equal facility, there appears to be much room for improvement in classifying special forms and in developing algorithms that are tailored to their practical processing. In view of these constraints and contingencies, my focus shifts to the tasks of approximation and interpretation that support intuition, especially in dealing with the natural kinds of differential propositions that arise in applications, and in the effort to understand, in succinct and adaptive forms, their dynamic implications. In the absence of direct insights, these tasks are partially carried out by forging analogies with the familiar situations and customary routines of ordinary calculus. But the indirect approach, going by way of specious analogy and intuitive habit, forces us to remain on guard against the circumstance that occurs when the word "forging" takes on its shadier nuance, indicting the constant risk of a counterfeit in the proportion. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D18 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | I would have preferred to be enveloped in words, | borne way beyond all possible beginnings. | | Michel Foucault, 'The Discourse on Language', [Fou, 215] Back to the Beginning: Some Exemplary Universes To anchor our understanding of differential logic, let us look at how the various concepts apply in the simplest possible concrete cases, where the initial dimension is only 1 or 2. In spite of the obvious simplicity of these cases, it is possible to observe how central difficulties of the subject begin to arise already at this stage. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D19 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o A One-Dimensional Universe | There was never any more inception than there is now, | Nor any more youth or age than there is now; | And will never be any more perfection than there is now, | Nor any more heaven or hell than there is now. | | Walt Whitman, Leaves of Grass, [Whi, 28] Let !X! = {x_1} = {A} be an alphabet that represents one boolean variable or a single logical feature. In this example I am using the capital letter "A" in a more usual informal way, to name a feature and not a space, at variance with my formerly stated formal conventions. At any rate, the basis element A = x_1 may be interpreted as a simple proposition or a coordinate projection A = x_1 : B^1 -:> B. The space X = <|A|> = {(A), A} of points (cells, vectors, interpretations) has cardinality 2^n = 2^1 = 2 and is isomorphic to B = {0, 1}. Moreover, X may be identified with the set of singular propositions {x : B ::> B}. The space of linear propositions X* = {hom : B ++> B} = {0, A} is algebraically dual to X and also has cardinality 2. Here, "0" is interpreted as denoting the constant function 0 : B -> B, amounting to the linear proposition of rank 0, while A is the linear proposition of rank 1. Last but not least we have the positive propositions {pos : B oo> B} = {A, 1}, of rank 1 and 0, respectively, where "1" is understood as denoting the constant function 1 : B -> B. In sum, there are 2^2^n = 2^2^1 = 4 propositions altogether in the universe of discourse, comprising the set X^ = {f : X -> B} = {0, (A), A, 1} ~=~ (B -> B). The first order differential extension of !X! is E!X! = {x_1, dx_1} = {A, dA}. If the feature "A" is understood as applying to some object or state, then the feature "dA" may be interpreted as an attribute of the same object or state that says that it is changing "significantly" with respect to the property A, or that it has an "escape velocity" with respect to the state A. In practice, differential features acquire their logical meaning through a class of "temporal inference rules". For example, relative to a frame of observation that is left implicit for now, one is permitted to make the following sorts of inference: from the fact that A and dA are true at a given moment one may infer that (A) will be true in the next moment of observation. Altogether in the present instance, there is the fourfold scheme of inference that is shown below: o-------------------------------------------------o | | | From (A) & (dA) infer (A) next. | | | | From (A) & dA infer A next. | | | | From A & (dA) infer A next. | | | | From A & dA infer (A) next. | | | o-------------------------------------------------o It might be thought that we need to bring in an independent time variable at this point, but an insight of fundamental importance appears to be that the idea of process is more basic than the notion of time. A time variable is actually a reference to a "clock", that is, a canonical or a convenient process that is established or accepted as a standard of measurement, but in essence no different than any other process. This raises the question of how different subsystems in a more global process can be brought into comparison, and what it means for one process to serve the function of a local standard for others. But these inquiries only wrap up puzzles in further riddles, and are obviously too involved to be handled at our current level of approximation. | The clock indicates the moment . . . . but what does | eternity indicate? | | Walt Whitman, 'Leaves of Grass', [Whi, 79] Observe that the secular inference rules, used by themselves, involve a loss of information, since nothing in them can tell us whether the momenta {(dA), dA} are preserved or changed in the next instance. In order to know this, we would have to determine d^2.A, and so on, pursuing an infinite regress. Ultimately, in order to rest with a finitely determinate system, it is necessary to make an infinite assumption, for example, that d^k.A = 0 for all k greater than some fixed value M. Another way to escape the regress is through the provision of a dynamic law, in typical form making higher order differentials dependent on lower degrees and estates. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D20 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Example 1. A Square Rigging | Urge and urge and urge, | Always the procreant urge of the world. | | Walt Whitman, 'Leaves of Grass', [Whi, 28] By way of example, suppose that we are given the initial condition A = dA and the law d^2.A = (A). Then, since "A = dA" <=> "A dA or (A)(dA)", we may infer two possible trajectories, as displayed in Table 11. In either of these cases, the state A (dA)(d^2.A) is a stable attractor or a terminal condition for both starting points. Table 11. A Pair of Commodious Trajectories o---------o-------------------o-------------------o | Time | Trajectory 1 | Trajectory 2 | o---------o-------------------o-------------------o | | | | | 0 | A dA (d^2.A) | (A) (dA) d^2.A | | | | | | 1 | (A) dA d^2.A | (A) dA d^2.A | | | | | | 2 | A (dA) (d^2.A) | A (dA) (d^2.A) | | | | | | 3 | A (dA) (d^2.A) | A (dA) (d^2.A) | | | | | | 4 | " " " | " " " | | | | | o---------o-------------------o-------------------o Because the initial space X = <|A|> is one-dimensional, we can easily fit the second order extension E^2.X = <|A, dA, d^2.A|> within the compass of a single venn diagram, charting the couple of converging trajectories as shown in Figure 12. o-------------------------------------------------o | E^2.X | | | | o-------------o | | / \ | | / A \ | | / \ | | / ->- \ | | o / \ o | | | \ / | | | | -o- | | | | ^ | | | o---o---------o | o---------o---o | | / \ \|/ / \ | | / \ o | / \ | | / \ | /|\ / \ | | / \ | / | \ / \ | | o o-|-o--|--o---o o | | | | | | | | | | | ---->o<----o | | | | | | | | | o dA o o d^2.A o | | \ \ / / | | \ \ / / | | \ o / | | \ / \ / | | o-------------o o-------------o | | | | | o-------------------------------------------------o Figure 12. The Anchor If we eliminate from view the regions of E^2.X that are ruled out by the dynamic law d^2.A = (A), then what remains is the quotient structure that is shown in Figure 13. This picture makes it easy to see that the dynamically allowable portion of the universe is partitioned between the properties A and d^2.A. As it happens, this fact might have been expressed "right off the bat" by an equivalent formulation of the differential law, one that uses the exclusive disjunction to state the law as (A, d^2.A). o-------------------------------------------------o | | | ->- | | / \ | | \ / | | o-------------o -o- | | / \ ^ | | / dA \/ A | | / /\ | | / / \ | | o o / o | | | \ / | | | | \ / | | o------------|-------\-------/-------|------------o | | \ / | | | | \ / | | | o v / o | | \ o / | | \ ^ / | | \ | / d^2.A | | \ | / | | o------|------o | | | | | | | | o | | | o-------------------------------------------------o Figure 13. The Tiller What we have achieved in this example is to give a differential description of a simple dynamic process. In effect, we did this by embedding a directed graph, which can be taken to represent the state transitions of a finite automaton, in a dynamically allotted quotient structure that is created from a boolean lattice or an n-cube by nullifying all of the regions that the dynamics outlaws. With growth in the dimensions of our contemplated universes, it becomes essential, both for human comprehension and for computer implementation, that the dynamic structures of interest to us be represented not actually, by acquaintance, but virtually, by description. In our present study, we are using the language of propositional calculus to express the relevant descriptions, and to comprehend the structure that is implicit in the subsets of a n-cube without necessarily being forced to actualize all of its points. One of the reasons for engaging in this kind of extremely reduced, but explicitly controlled case study is to throw light on the general study of languages, formal and natural, in their full array of syntactic, semantic, and pragmatic aspects. Propositional calculus is one of the last points of departure where we can view these three aspects interacting in a non-trivial way without being immediately and totally overwhelmed by the complexity they generate. Often this complexity causes investigators of formal and natural languages to adopt the strategy of focusing on a single aspect and to abandon all hope of understanding the whole, whether it's the still living natural language or the dynamics of inquiry that lies crystallized in formal logic. From the perspective that I find most useful here, a language is a syntactic system that is designed or evolved in part to express a set of descriptions. When the explicit symbols of a language have extensions in its object world that are actually infinite, or when the implicit categories and generative devices of a linguistic theory have extensions in its subject matter that are potentially infinite, then the finite characters of terms, statements, arguments, grammars, logics, and rhetorics force an excess of intension to reside in all these symbols and functions, across the spectrum from the object language to the metalinguistic uses. In the aphorism from W. von Humboldt that Chomsky often cites, for example, in [Cho86, 30] and [Cho93, 49], language requires "the infinite use of finite means". This is necessarily true when the extensions are infinite, when the referential symbols and grammatical categories of a language possess infinite sets of models and instances. But it also voices a practical truth when the extensions, though finite at every stage, tend to grow at exponential rates. This consequence of dealing with extensions that are "practically infinite" becomes crucial when one tries to build neural network systems that learn, since the learning competence of any intelligent system is limited to the objects and domains that it is able to represent. If we want to design systems that operate intelligently with the full deck of propositions dealt by intact universes of discourse, then we must supply them with succinct representations and efficient transformations in this domain. Furthermore, in the project of constructing inquiry driven systems, we find ourselves forced to contemplate the level of generality that is embodied in propositions, because the dynamic evolution of these systems is driven by the measurable discrepancies that occur among their expectations, intentions, and observations, and because each of these subsystems or components of knowledge constitutes a propositional modality that can take on the fully generic character of an empirical summary or an axiomatic theory. A compression scheme by any other name is a symbolic representation, and this is what the differential extension of propositional calculus, through all of its many universes of discourse, is intended to supply. Why is this particular program of mental calisthenics worth carrying out in general? By providing a uniform logical medium for describing dynamic systems we can make the task of understanding complex systems much easier, both in looking for invariant representations of individual cases and in finding points of comparison among diverse structures that would otherwise appear as isolated systems. All of this goes to facilitate the search for compact knowledge and to adapt what is learned from individual cases to the general realm. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D21 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Back to the Feature | I guess it must be the flag of my disposition, out of hopeful | green stuff woven. | | Walt Whitman, 'Leaves of Grass', [Whi, 31] Let us assume that the sense intended for differential features is well enough established in the intuition, for now, that I may continue with outlining the structure of the differential extension [E!X!] = [A, dA]. Over the extended alphabet E!X! = {x_1, dx_1} = {A, dA}, of cardinality 2^n = 2, we generate the set of points, EX, of cardinality 2^2n = 4, that bears the following chain of equivalent descriptions: EX = <|A, dA|> = {(A), A} x {(dA), dA} = {(A)(dA), (A) dA, A (dA), A dA}. The space EX may be assigned the mnemonic type B x D, which is really no different than B x B = B^2. An individual element of EX may be regarded as a "disposition at a point" or a "situated direction", in effect, a singular mode of change occurring at a single point in the universe of discourse. In applications, the modality of this change can be interpreted in various ways, for example, as an expectation, an intention, or an observation with respect the behavior of a system. To complete the construction of the extended universe of discourse EX% = [x_1, dx_1] = [A, dA], one must add the set of differential propositions EX^ = {g : EX -> B} ~=~ (B x D -> B) to the set of dispositions in EX. There are 2^2^2n = 16 propositions in EX^, as detailed in Table 14. Table 14. Differential Propositions o-------o--------o---------o-----------o-------------------o----------o | | A : 1 1 0 0 | | | | | | dA : 1 0 1 0 | | | | o-------o--------o---------o-----------o-------------------o----------o | | | | | | | | f_0 | g_0 | 0 0 0 0 | () | False | 0 | | | | | | | | o-------o--------o---------o-----------o-------------------o----------o | | | | | | | | | g_1 | 0 0 0 1 | (A)(dA) | Neither A nor dA | ~A & ~dA | | | | | | | | | | g_2 | 0 0 1 0 | (A) dA | Not A but dA | ~A & dA | | | | | | | | | | g_4 | 0 1 0 0 | A (dA) | A but not dA | A & ~dA | | | | | | | | | | g_8 | 1 0 0 0 | A dA | A and dA | A & dA | | | | | | | | o-------o--------o---------o-----------o-------------------o----------o | | | | | | | | f_1 | g_3 | 0 0 1 1 | (A) | Not A | ~A | | | | | | | | | f_2 | g_12 | 1 1 0 0 | A | A | A | | | | | | | | o-------o--------o---------o-----------o-------------------o----------o | | | | | | | | | g_6 | 0 1 1 0 | (A, dA) | A not equal to dA | A + dA | | | | | | | | | | g_9 | 1 0 0 1 | ((A, dA)) | A equal to dA | A = dA | | | | | | | | o-------o--------o---------o-----------o-------------------o----------o | | | | | | | | | g_5 | 0 1 0 1 | (dA) | Not dA | ~dA | | | | | | | | | | g_10 | 1 0 1 0 | dA | dA | dA | | | | | | | | o-------o--------o---------o-----------o-------------------o----------o | | | | | | | | | g_7 | 0 1 1 1 | (A dA) | Not both A and dA | ~A v ~dA | | | | | | | | | | g_11 | 1 0 1 1 | (A (dA)) | Not A without dA | A => dA | | | | | | | | | | g_13 | 1 1 0 1 | ((A) dA) | Not dA without A | A <= dA | | | | | | | | | | g_14 | 1 1 1 0 | ((A)(dA)) | A or dA | A v dA | | | | | | | | o-------o--------o---------o-----------o-------------------o----------o | | | | | | | | f_3 | g_15 | 1 1 1 1 | (()) | True | 1 | | | | | | | | o-------o--------o---------o-----------o-------------------o----------o Aside from changing the names of variables and shuffling the order of rows, this Table follows the format that was used previously for boolean functions of two variables. The rows are grouped to reflect natural similarity classes among the propositions. In a future discussion, these classes will be given additional explanation and motivation as the orbits of a certain transformation group acting on the set of 16 propositions. Notice that four of the propositions, in their logical expressions, resemble those given in the table for X^. Thus the first set of propositions {f_i} is automatically embedded in the present set {g_j}, and the corresponding inclusions are indicated at the far left margin of the table. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D22 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Tacit Extensions | I would really like to have slipped imperceptibly into this lecture, as | into all the others I shall be delivering, perhaps over the years ahead. | | Michel Foucault, 'The Discourse on Language', [Fou, 215] Strictly speaking, however, there is a subtle distinction in type between the function f_i : X -> B and the corresponding function g_j : EX -> B, even though they share the same logical expression. Being human, we insist on preserving all the aesthetic delights afforded by the abstractly unified form of the "cake" while giving up none of the diverse contents that its substantive consummation can provide. In short, we want to maintain the logical equivalence of expressions that represent the same proposition, while appreciating the full diversity of that proposition's functional and typical representatives. Both perspectives, and all the levels of abstraction extending through them, have their reasons, as will develop in time. Because this special circumstance points up an important general theme, it is a good idea to discuss it more carefully. Whenever there arises a situation like this, where one alphabet !X! is a subset of another alphabet !Y!, then we say that any proposition f : <|!X!|> -> B has a "tacit extension" to a proposition !e!f : <|!Y!|> -> B, and that the space (<|!X!|> -> B) has an "automatic embedding" within the space (<|!Y!|> -> B). The extension is defined in such a way that !e!f puts the same constraint on the variables of X that are contained in Y as the proposition f initially did, while it puts no constraint on the variables of Y outside of X, in effect, conjoining the two constraints. If the variables in question are indexed as !X! = {x_1, ..., x_n} and !Y! = {x_1, ..., x_n, ..., x_n+k}, then the definition of the tacit extension from !X! to !Y! may be expressed in the form of an equation: !e!f(x_1, ..., x_n, ..., x_n+k) = f(x_1, ..., x_n). On formal occasions, such as the present context of definition, the tacit extension from !X! to !Y! is explicitly symbolized by the operator !e! : (<|!X!|> -> B) -> (<|!Y!|> -> B), where the appropriate alphabets !X! and !Y! are understood from context, but normally one may leave the "!e!" silent. Let's explore what this means for the present Example. Here, !X! = {A} and !Y! = E!X! = {A, dA}. For each of the propositions f_i over X, specifically, those whose expression e_i lies in the collection {0, (A), A, 1}, the tacit extension !e!f of f to EX can be phrased as a logical conjunction of two factors, f_i = e_i.!t!, where !t! is a logical tautology that uses all the variables of !Y! - !X!. Working in these terms, the tacit extensions !e!f of f to EX may be explicated as shown in Table 15. Table 15. Tacit Extensions of [A] to [A, dA] o---------------------------------------------------------------------o | | | 0 = 0 . ((dA), dA) = 0 | | | | (A) = (A) . ((dA), dA) = (A)(dA) + (A) dA | | | | A = A . ((dA), dA) = A (dA) + A dA | | | | 1 = 1 . ((dA), dA) = 1 | | | o---------------------------------------------------------------------o In its effect on the singular propositions over X, this analysis has an interesting interpretation. The tacit extension takes us from thinking about a particular state, like A or (A), to considering the collection of outcomes, the outgoing changes or the singular dispositions, that spring from that state. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D23 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Example 2. Drives and Their Vicissitudes | I open my scuttle at night and see the far-sprinkled systems, | And all I see, multiplied as high as I can cipher, edge but | the rim of the farther systems. | | Walt Whitman, 'Leaves of Grass', [Whi, 81] Before we leave the one-feature case let's look at a more substantial example, one that illustrates a general class of curves that can be charted through the extended feature spaces and that provides an opportunity to discuss a number of important themes concerning their structure and dynamics. Again, let !X! = {x_1} = {A}. In the discussion that follows I will consider a class of trajectories having the property that d^k.A = 0 for all k greater than some fixed m, and I indulge in the use of some picturesque terms that describe salient classes of such curves. Given the finite order condition, there is a highest order non-zero difference d^m.A exhibited at each point in the course of any determinate trajectory that one may wish to consider. With respect to any point of the corresponding orbit or curve let us call this highest order differential feature d^m.A the "drive" at that point. Curves of constant drive d^m.A are then referred to as "m^th gear curves". Scholium. The fact that a difference calculus can be developed for boolean functions is well known [Fuji], [Koh, sec. 8-4] and was probably familiar to Boole, who was an expert in difference equations before he turned to logic. And of course there is the strange but true story of how the Turin machines of the 1840's prefigured the Turing machines of the 1940's [Men, 225-297]. At the very outset of general purpose, mechanized computing we find that the motive power driving the Analytical Engine of Babbage, the kernel of an idea behind all of his wheels, was exactly his notion that difference operations, suitably trained, can serve as universal joints for any conceivable computation [M&M], [Mel, ch. 4]. Given this language, the particular Example that I take up here can be described as the family of 4^th gear curves through E^4.X = <|A, dA, d^2.A, d^3.A, d^4.A|>. These are the trajectories generated subject to the dynamic law d^4.A = 1, where it is understood in such a statement that all higher order differences are equal to 0. Since d^4.A and all higher d^k.A are fixed, the temporal or transitional conditions (initial, mediate, terminal -- transient or stable states) vary only with respect to their projections as points of E^3.X = <|A, dA, d^2.A, d^3.A|>. Thus, there is just enough space in a planar venn diagram to plot all of these orbits and to show how they partition the points of E^3.X. It turns out that there are exactly two possible orbits, of eight points each, as illustrated in Figures 16-a and 16-b. (NB. I leave it as an exercise for the reader to connect the dots in the second figure.) o-------------------------------------------------o | | | o | | / \ | | / \ | | / \ | | / \ | | o o | | / \ / \ | | / \ / \ | | / \ / \ | | / \ / \ | | o o o | | / \ / \ / \ | | / \ / \ / \ | | / \ / \ / \ | | / \ / \ / \ | | o 5 o 7 o o | | / \ ^| / \ ^| / \ / \ | | / \/ | / \/ | / \ / \ | | / /\ | / /\ | / \ / \ | | / / \|/ / \|/ \ / \ | | o 4<---|----/----|----3 o o | | |\ /|\ / /|\ ^ / \ /| | | | \ / | \/ / | \/ / \ / | | | | \ / | /\ / | /\ / \ / | | | | \ / v/ \ / |/ \ / \ / | | | | o 6 o | o o | | | | |\ / \ /| / \ /| | | | | | \ / \/ | / \ / | | | | | | \ / /\ | / \ / | | | | | d^0.A \ / / \|/ \ / d^1.A | | | o----+----o 2<---|----1 o----+----o | | | \ /|\ ^ / | | | | \ / | \/ / | | | | \ / | /\ / | | | | d^2.A \ / v/ \ / d^3.A | | | o---------o 0 o---------o | | \ / | | \ / | | \ / | | \ / | | o | | | o-------------------------------------------------o Figure 16-a. A Couple of Fourth Gear Orbits: 1 o-------------------------------------------------o | | | o | | / \ | | / \ | | / \ | | / \ | | o 0 o | | / \ / \ | | / \ / \ | | / \ / \ | | / \ / \ | | o 5 o 2 o | | / \ / \ / \ | | / \ / \ / \ | | / \ / \ / \ | | / \ / \ / \ | | o o o 6 o | | / \ / \ / \ / \ | | / \ / \ / \ / \ | | / \ / \ / \ / \ | | / \ / \ / \ / \ | | o o 7 o o 4 o | | |\ / \ / \ / \ /| | | | \ / \ / \ / \ / | | | | \ / \ / \ / \ / | | | | \ / \ / \ / \ / | | | | o o 3 o 1 o | | | | |\ / \ / \ /| | | | | | \ / \ / \ / | | | | | | \ / \ / \ / | | | | | d^0.A \ / \ / \ / d^1.A | | | o----+----o o o----+----o | | | \ / \ / | | | | \ / \ / | | | | \ / \ / | | | | d^2.A \ / \ / d^3.A | | | o---------o o---------o | | \ / | | \ / | | \ / | | \ / | | o | | | o-------------------------------------------------o Figure 16-b. A Couple of Fourth Gear Orbits: 2 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D24 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Example 2. Drives and Their Vicissitudes (concl.) With a little thought it is possible to devise an indexing scheme for the general run of dynamic states that allows for comparing universes of discourse that weigh in on different scales of observation. With this end in sight, let us index the states q in E^m.X with the dyadic rationals (or the binary fractions) in the half-open interval [0, 2). Formally and canonically, a state q_r is indexed by a fraction r = s/t whose denominator is the power of two t = 2^m and whose numerator is a binary numeral that is formed from the coefficients of state in a manner to be described next. The "differential coefficients" of the state q are just the values d^k.A(q), for k = 0 to m, where d^0.A is defined as being identical to A. To form the binary index d_0 . d_1 ... d_m of the state q the coefficient d^k.A(q) is read off as the binary digit d_k associated with the place value 2^(-k). Expressed by way of algebraic formulas, the rational index r of the state q can be given by the following equivalent formulations: o-------------------------------------------------------------------------------o | | | r(q) = Sum_k d_k . 2^(-k) = Sum_k d^k.A(q) . 2^(-k) | | | | = | | | | s(q)/t = (Sum_k d_k . 2^(m-k)) / 2^m = (Sum_k d^k.A(q) . 2^(m-k)) / 2^m | | | o-------------------------------------------------------------------------------o Applied to the example of fourth gear curves, this scheme results in the data of Tables 17-a and 17-b, which exhibit one period for each orbit. The states in each orbit are listed as ordered pairs <p_i, q_j>, where p_i may be read as a temporal parameter that indicates the present time of the state, and where j is the decimal equivalent of the binary numeral 's'. Informally and more casually, the Tables exhibit the states q_s as subscripted with the numerators of their rational indices, taking for granted the constant denominators of 2^m = 2^4 = 16. Within this set-up, the temporal successions of states can be reckoned as given by a kind of "parallel round-up rule". That is, if <d_k, d_(k+1)> is any pair of adjacent digits in the state index r, then the value of d_k in the next state is (d_k)' = d_k + d_(k+1). Table 17-a. A Couple of Orbits in Fourth Gear: Orbit 1 o---------o---------o---------o---------o---------o---------o---------o | Time | State | A | dA | | | | | p_i | q_j | d^0.A | d^1.A | d^2.A | d^3.A | d^4.A | o---------o---------o---------o---------o---------o---------o---------o | | | | | p_0 | q_01 | 0. 0 0 0 1 | | | | | | p_1 | q_03 | 0. 0 0 1 1 | | | | | | p_2 | q_05 | 0. 0 1 0 1 | | | | | | p_3 | q_15 | 0. 1 1 1 1 | | | | | | p_4 | q_17 | 1. 0 0 0 1 | | | | | | p_5 | q_19 | 1. 0 0 1 1 | | | | | | p_6 | q_21 | 1. 0 1 0 1 | | | | | | p_7 | q_31 | 1. 1 1 1 1 | | | | | o---------o---------o---------o---------o---------o---------o---------o Table 17-b. A Couple of Orbits in Fourth Gear: Orbit 2 o---------o---------o---------o---------o---------o---------o---------o | Time | State | A | dA | | | | | p_i | q_j | d^0.A | d^1.A | d^2.A | d^3.A | d^4.A | o---------o---------o---------o---------o---------o---------o---------o | | | | | p_0 | q_25 | 1. 1 0 0 1 | | | | | | p_1 | q_11 | 0. 1 0 1 1 | | | | | | p_2 | q_29 | 1. 1 1 0 1 | | | | | | p_3 | q_07 | 0. 0 1 1 1 | | | | | | p_4 | q_09 | 0. 1 0 0 1 | | | | | | p_5 | q_27 | 1. 1 0 1 1 | | | | | | p_6 | q_13 | 0. 1 1 0 1 | | | | | | p_7 | q_23 | 1. 0 1 1 1 | | | | | o---------o---------o---------o---------o---------o---------o---------o o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D25 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | It is understandable that an engineer should be completely absorbed in his | speciality, instead of pouring himself out into the freedom and vastness | of the world of thought, even though his machines are being sent off | to the ends of the earth; for he no more needs to be capable of | applying to his own personal soul what is daring and new | in the soul of his subject than a machine is in fact | capable of applying to itself the differential | calculus on which it is based. The same | thing cannot, however, be said about | mathematics; for here we have | the new method of thought, | pure intellect, the very | well-spring of the times, | the 'fons et origo' of an | unfathomable transformation. | | Robert Musil, 'The Man Without Qualities', [Mus, 39] Transformations of Discourse In this section we take up the general study of logical transformations, or maps that relate one universe of discourse to another. In many ways, and especially as applied to the subject of intelligent dynamic systems, my argument develops the antithesis of the statement just quoted. Along the way, if incidental to my ends, I hope this essay can pose a fittingly irenic epitaph to the frankly ironic epigram I have inscribed at its head. My goal in this section is to answer a single question: What is a propositional tangent functor? In other words, my aim is to develop a clear conception of what manner of thing would pass in the logical realm for a genuine analogue of the tangent functor, an object conceived to generalize as far as possible in the abstract terms of category theory the ordinary notions of functional differentiation and the all too familiar operations of taking derivatives. As a first step I discuss the kinds of transformations that we already know as "extensions" and "projections", and I use these special cases to illustrate several different styles of logical and visual representation that will figure heavily in the sequel. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D26 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | And, despite the care which she took to look behind her at every moment, | she failed to see a shadow which followed her like her own shadow, which | stopped when she stopped, which started again when she did, and which made | no more noise than a well-conducted shadow should. | | Gaston Leroux, 'The Phantom of the Opera', [Ler, 126] Foreshadowing Transformations: Extensions and Projections of Discourse Many times in our discussion we have occasion to place one universe of discourse in the context of a larger universe of discourse. An embedding of the general type [!X!] -> [!Y!] is implied any time that we make use of one alphabet !X! that happens to be included in another alphabet !Y!. When we are discussing differential issues we usually have in mind that the extended alphabet !Y! has a special construction or a specific lexical relation with respect to the initial alphabet !X!, one that is marked by characteristic types of accents, indices, or inflected forms. Extension from 1 to 2 Dimensions Figure 18-a lays out the "angular form" of venn diagram for universes of 1 and 2 dimensions, indicating the embedding map of type B^1 -> B^2 and detailing the coordinates that are associated with individual cells. Because all points, cells, or logical interpretations are represented as connected geometric areas, we can say that these pictures provide us with an "areal view" of each universe of discourse. o-----------------------------------------------------------o | | | o o | | / \ / \ | | / \ / \ | | / \ / \ | | / \ / \ | | / o o 1 1 o | | / / \ / \ / \ | | / / \ / \ / \ | | / 1 / \ / \ / \ | | / / \ !e! / \ / \ | | o / o ----> o 1 0 o 0 1 o | | |\ / / |\ / \ /| | | | \ / 0 / | \ / \ / | | | | \ / / | \ / \ / | | | |x_1\ / / |x_1\ / \ /x_2| | | o----o / o----o 0 0 o----o | | \ / \ / | | \ / \ / | | \ / \ / | | \ / \ / | | o o | | | o-----------------------------------------------------------o Figure 18-a. Extension from 1 to 2 Dimensions: Areal Figure 18-b shows the differential extension from X% = [x] to EX% = [x, dx] in a "bundle of boxes" form of venn diagram. As awkward as it may seem at first, this type of picture is often the most natural and the most easily available representation when we want to conceptualize the localized information or momentary knowledge of an intelligent dynamic system. It gives a ready picture of a "proposition at a point", in the present instance, of a proposition about changing states which is itself associated with a particular dynamic state of a system. It is easy to see how this application might be extended to conceive of more general types of instantaneous knowledge that are possessed by a system. o-----------------------------o o-------------------o | | | | | | | o-------o | | o---------o | | / \ | | / \ | | o o | | / o------------------------| | dx | | | / \ | | o o | | / \ | | \ / | | o o | | o-------o | | | | | | | | | | | o-------------------o | | x | | | | | | o-------------------o | | | | | | | o o | | o-------o | | \ / | | / \ | | \ / | | o o | | \ / o------------| | dx | | | \ / | | o o | | o---------o | | \ / | | | | o-------o | | | | | o-----------------------------o o-------------------o Figure 18-b. Extension from 1 to 2 Dimensions: Bundle Figure 18-c shows the same extension in a "compact" style of venn diagram, where the differential features at each position are represented by arrows extending from that position that cross or do not cross, as the case may be, the corresponding feature boundaries. o-----------------------------------------------------------o | | | | | o-----------------o | | / o \ | | / (dx) / \ \ dx | | / v o--------------------->o | | / \ / \ | | / o \ | | o o | | | | | | | | | | | x | (x) | | | | | | | | | | o o | | \ / o | | \ / / \ | | \ o<---------------------o v | | \ / dx \ / (dx) | | \ / o | | o-----------------o | | | | | o-----------------------------------------------------------o Figure 18-c. Extension from 1 to 2 Dimensions: Compact Figure 18-d compresses the picture of the differential extension even further, yielding a directed graph or "digraph" form of representation. (Notice that my definition of a digraph allows for loops or "slings" at individual points, in addition to arcs or "arrows" between the points.) o-----------------------------------------------------------o | | | | | dx | | .--. .---------->----------. .--. | | | \ / \ / | | | (dx) ^ @ x (x) @ v (dx) | | | / \ / \ | | | *--* *----------<----------* *--* | | dx | | | | | o-----------------------------------------------------------o Figure 18-d. Extension from 1 to 2 Dimensions: Digraph o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D27 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Extension from 2 to 4 Dimensions Figure 19-a lays out the "areal view" or the "angular form" of venn diagram for universes of 2 and 4 dimensions, indicating the embedding map of type B^2 -> B^4. In many ways these pictures are the best kind there is, giving full canvass to an ideal vista. Their style allows the clearest, the fairest, and the plainest view that we can form of a universe of discourse, affording equal representation to all dispositions and maintaining a balance with respect to ordinary and differential features. If only we could extend this view! Unluckily, an obvious difficulty beclouds this prospect, and that is how precipitately we run into the limits of our plane and visual intuitions. Even within the scope of the spare few dimensions that we have scanned up to this point subtle discrepancies have crept in already. The circumstances that bind us and the frameworks that block us, the flat distortion of the planar projection and the inevitable ineffability that precludes us from wrapping its rhomb figure into rings around a torus, all of these factors disguise the underlying but true connectivity of the universe of discourse. o-------------------------------------------------------------------------------o | | | o o | | / \ / \ | | / \ / \ | | / \ / \ | | / \ o 1100 o | | / \ / \ / \ | | / \ / \ / \ | | / \ !e! / \ / \ | | o 1 1 o ----> o 1101 o 1110 o | | / \ / \ / \ / \ / \ | | / \ / \ / \ / \ / \ | | / \ / \ / \ / \ / \ | | / \ / \ o 1001 o 1111 o 0110 o | | / \ / \ / \ / \ / \ / \ | | / \ / \ / \ / \ / \ / \ | | / \ / \ / \ / \ / \ / \ | | o 1 0 o 0 1 o o 1000 o 1011 o 0111 o 0100 o | | |\ / \ /| |\ / \ / \ / \ /| | | | \ / \ / | | \ / \ / \ / \ / | | | | \ / \ / | | \ / \ / \ / \ / | | | | \ / \ / | | o 1010 o 0011 o 0101 o | | | | \ / \ / | | |\ / \ / \ /| | | | | \ / \ / | | | \ / \ / \ / | | | | | x_1 \ / \ / x_2 | |x_1| \ / \ / \ / |x_2| | | o-------o 0 0 o-------o o---+---o 0010 o 0001 o---+---o | | \ / | \ / \ / | | | \ / | \ / \ / | | | \ / | x_3 \ / \ / x_4 | | | \ / o-------o 0000 o-------o | | \ / \ / | | \ / \ / | | \ / \ / | | o o | | | o-------------------------------------------------------------------------------o Figure 19-a. Extension from 2 to 4 Dimensions: Areal Figure 19-b shows the differential extension from U% = [u, v] to EU% = [u, v, du, dv] in the "bundle of boxes" form of venn diagram. o-----------------------------o | o-----o o-----o | | / \ / \ | | / o \ | | / / \ \ | | o o o o | @ | du | | dv | | /| o o o o | / | \ \ / / | / | \ o / | / | \ / \ / | / | o-----o o-----o | / o-----------------------------o / o-----------------------------------------/---o o-----------------------------o | / | | o-----o o-----o | | @ | | / \ / \ | | o---------o o---------o | | / o \ | | / \ / \ | | / / \ \ | | / o \ | | o o o o | | / / \ @-------\-----------@ | du | | dv | | | / / @ \ \ | | o o o o | | / / \ \ \ | | \ \ / / | | / / \ \ \ | | \ o / | | o o \ o o | | \ / \ / | | | | \| | | | o-----o o-----o | | | | | | | o-----------------------------o | | u | |\ v | | | | | | \ | | o-----------------------------o | | | | \ | | | o-----o o-----o | | o o o \ o | | / \ / \ | | \ \ / \ / | | / o \ | | \ \ / \ / | | / / \ \ | | \ \ / \ / | | o o o o | | \ @-----\-/-----------\-------------@ | du | | dv | | | \ o / | | o o o o | | \ / \ / \ | | \ \ / / | | o---------o o---------o \ | | \ o / | | \ | | \ / \ / | | \ | | o-----o o-----o | o-----------------------------------------\---o o-----------------------------o \ \ o-----------------------------o \ | o-----o o-----o | \ | / \ / \ | \ | / o \ | \ | / / \ \ | \| o o o o | @ | du | | dv | | | o o o o | | \ \ / / | | \ o / | | \ / \ / | | o-----o o-----o | o-----------------------------o Figure 19-b. Extension from 2 to 4 Dimensions: Bundle As dimensions increase, this factorization of the extended universe along the lines that are marked out by the bundle picture begins to look more and more like a practical necessity. But whenever we use a propositional model to address a real situation in the context of nature we need to remain aware that this articulation into factors, affecting our description, may be wholly artificial in nature and cleave to nothing, no joint in nature, nor any juncture in time to be in or out of joint. Figure 19-c illustrates the extension from 2 to 4 dimensions in the "compact" style of venn diagram. Here, just the changes with respect to the center cell are shown. o---------------------------------------------------------------------o | | | | | o-------------------o o-------------------o | | / \ / \ | | / o \ | | / / \ \ | | / / \ \ | | / / \ \ | | / / \ \ | | / / \ \ | | o o (du).(dv) o o | | | | -->-- | | | | | | \ / | | | | | dv .(du) | \ / | du .(dv) | | | | u <---------------@---------------> v | | | | | | | | | | | | | | | | | | | | | | | | o o | o o | | \ \ | / / | | \ \ | / / | | \ \ | / / | | \ \ | / / | | \ \|/ / | | \ | / | | \ /|\ / | | o-------------------o | o-------------------o | | | | | du . dv | | | | | V | | | o---------------------------------------------------------------------o Figure 19-c. Extension from 2 to 4 Dimensions: Compact Figure 19-d gives the "digraph" form of representation for the differential extension U% -> EU%, where the 4 nodes marked "@" are the cells uv, u(v), (u)v, (u)(v), respectively, and where a 2-headed arc counts as two arcs of the differential digraph. o-----------------------------------------------------------o | | | .->-. | | | | | | * * | | \ / | | .-->--@--<--. | | / / \ \ | | / / \ \ | | / . . \ | | / | | \ | | / | | \ | | / | | \ | | . | | . | | | | | | | | v | | v | | .--. | .---------->----------. | .--. | | | \|/ | | \|/ | | | ^ @ ^ v @ v | | | /|\ | | /|\ | | | *--* | *----------<----------* | *--* | | ^ | | ^ | | | | | | | | * | | * | | \ | | / | | \ | | / | | \ | | / | | \ . . / | | \ \ / / | | \ \ / / | | *-->--@--<--* | | / \ | | . . | | | | | | *-<-* | | | o-----------------------------------------------------------o Figure 19-d. Extension from 2 to 4 Dimensions: Digraph o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D28 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | And as imagination bodies forth | The forms of things unknown, the poet's pen | Turns them to shapes, and gives to airy nothing | A local habitation and a name. | | 'A Midsummer Night's Dream', 5.1.18 Thematization of Functions: And a Declaration of Independence for Variables In the representation of propositions as functions it is possible to notice different degrees of explicitness in the way their functional character is symbolized. To indicate what I mean by this, the next series of Figures illustrates a set of graphic conventions that will be put to frequent use in the remainder of this discussion, both to mark the relevant distinctions and to help us convert between related expressions at different levels of explicitness in their functionality. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D29 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Thematization: Venn Diagrams | The known universe has one complete lover and that is the greatest poet. | He consumes an eternal passion and is indifferent which chance happens | and which possible contingency of fortune or misfortune and persuades | daily and hourly his delicious pay. | | Walt Whitman, 'Leaves of Grass', [Whi, 11-12] Figure 20-i traces the first couple of steps in this order of "thematic" progression, that will gradually run the gamut through a complete series of degrees of functional explicitness in the expression of logical propositions. The first venn diagram represents a situation where the function is indicated by a shaded figure and a logical expression. At this stage one may be thinking of the proposition only as expressed by a formula in a particular language and its content only as a subset of the universe of discourse, as when one considers the proposition u.v in [u, v]. The second venn diagram depicts a situation in which two significant steps have been taken. First, one has taken the trouble to give the proposition u.v a distinctive functional name "J". Second, one has come to think explicitly about the target domain that contains the functional values of J, as when one writes J : <|u, v|> -> B. o-------------------------------o o-------------------------------o | | | | | o-----o o-----o | | o-----o o-----o | | / \ / \ | | / \ / \ | | / o \ | | / o \ | | / /`\ \ | | / /`\ \ | | o o```o o | | o o```o o | | | u |```| v | | | | u |```| v | | | o o```o o | | o o```o o | | \ \`/ / | | \ \`/ / | | \ o / | | \ o / | | \ / \ / | | \ / \ / | | o-----o o-----o | | o-----o o-----o | | | | | o-------------------------------o o-------------------------------o \ / \ / \ / u v \ J / \ / \ / \ / \ / o Figure 20-i. Thematization of Conjunction (Stage 1) In Figure 20-ii the proposition J is viewed explicitly as a transformation from one universe of discourse to another. o-------------------------------o o-------------------------------o | | | | | o-----o o-----o | | o-----o o-----o | | / \ / \ | | / \ / \ | | / o \ | | / o \ | | / /`\ \ | | / /`\ \ | | o o```o o | | o o```o o | | | u |```| v | | | | u |```| v | | | o o```o o | | o o```o o | | \ \`/ / | | \ \`/ / | | \ o / | | \ o / | | \ / \ / | | \ / \ / | | o-----o o-----o | | o-----o o-----o | | | | | o-------------------------------o o-------------------------------o \ / \ / \ / \ / \ / \ J / \ / \ / \ / \ / o----------\---------/----------o o----------\---------/----------o | \ / | | \ / | | \ / | | \ / | | o-----@-----o | | o-----@-----o | | /`````````````\ | | /`````````````\ | | /```````````````\ | | /```````````````\ | | /`````````````````\ | | /`````````````````\ | | o```````````````````o | | o```````````````````o | | |```````````````````| | | |```````````````````| | | |```````` J ````````| | | |```````` x ````````| | | |```````````````````| | | |```````````````````| | | o```````````````````o | | o```````````````````o | | \`````````````````/ | | \`````````````````/ | | \```````````````/ | | \```````````````/ | | \`````````````/ | | \`````````````/ | | o-----------o | | o-----------o | | | | | | | | | o-------------------------------o o-------------------------------o J = u v x = J<u, v> Figure 20-ii. Thematization of Conjunction (Stage 2) In the first venn diagram the name that is assigned to a composite proposition, function, or region in the source universe is delegated to a simple feature in the target universe. This can result in a single character or term exceeding the responsibilities it can carry off well. Allowing the name of a function J : <|u, v|> -> B to serve as the name of its dependent variable J : B does not mean that one has to confuse a function with any of its values, but it does put one at risk for a number of obvious problems, and we should not be surprised, on numerous and limiting occasions, when quibbling arises from the attempts of a too original syntax to serve these two masters. The second venn diagram circumvents these difficulties by introducing a new variable name for each basic feature of the target universe, as when one writes J : <|u, v|> -> <|x|> and thereby assigns a concrete type <|x|> to the abstract codomain B. To make this induction of variables more formal one can append subscripts, as in x_J, to indicate the origin or the derivation of these parvenu characters. However, it is not always convenient to keep inventing new variable names in this way. For use at these times, I introduce a lexical operator "¢", read "cents" or "obelus", that converts a function name into a variable name. For example, one may think of x = x_J = ¢(J) = J¢ = J^¢ as "the cache variable of J", "J circumscript", "J made circumstantial", or "J considered as a contingent variable". In Figure 20-iii we arrive at a stage where the functional equations, J = u.v and x = u.v, are regarded as propositions in their own right, reigning in and ruling over 3-feature universes of discourse, [u, v, J] and [u, v, x], respectively. Subject to the cautions already noted, the function name "J" can be reinterpreted as the name of a feature J^¢, and the equation J = u.v can be read as the logical equivalence ((J, u v)). To give it a generic name let us call this newly expressed, collateral proposition the "thematization" or the "thematic extension" of the original proposition J. o-------------------------------o o-------------------------------o | | |```````````````````````````````| | | |````````````o-----o````````````| | | |```````````/ \```````````| | | |``````````/ \``````````| | | |`````````/ \`````````| | | |````````/ \````````| | J | |```````o x o```````| | | |```````| |```````| | | |```````| |```````| | | |```````| |```````| | o-----o o-----o | |```````o-----o o-----o```````| | / \ / \ | |``````/`\ \ / /`\``````| | / o \ | |`````/```\ o /```\`````| | / /`\ \ | |````/`````\ /`\ /`````\````| | / /```\ \ | |```/```````\ /```\ /```````\```| | o o`````o o | |``o`````````o-----o`````````o``| | | u |`````| v | | |``|`````````| |`````````|``| o--o---------o-----o---------o--o |``|``` u ```| |``` v ```|``| |``|`````````| |`````````|``| |``|`````````| |`````````|``| |``o`````````o o`````````o``| |``o`````````o o`````````o``| |```\`````````\ /`````````/```| |```\`````````\ /`````````/```| |````\`````````\ /`````````/````| |````\`````````\ /`````````/````| |`````\`````````o`````````/`````| |`````\`````````o`````````/`````| |``````\```````/`\```````/``````| |``````\```````/`\```````/``````| |```````o-----o```o-----o```````| |```````o-----o```o-----o```````| |```````````````````````````````| |```````````````````````````````| o-------------------------------o o-------------------------------o \ / \ / J = u v \ / \ !j! / \ / !j! = (( x , u v )) \ / \ / \ / @ Figure 20-iii. Thematization of Conjunction (Stage 3) The first venn diagram represents the thematization of the conjunction J with shading in the appropriate regions of the universe [u, v, J]. Also, it illustrates a quick way of constructing a thematic extension. First, draw a line, in practice or the imagination, that bisects every cell of the original universe, placing half of each cell under the aegis of the thematized proposition and the other half under its antithesis. Next, on the scene where the theme applies leave the shade wherever it lies, and off the stage, where it plays otherwise, stagger the pattern in a harlequin guise. In the final venn diagram of this sequence the thematic progression comes full circle and completes one round of its development. The ambiguities that were occasioned by the changing role of the name "J" are resolved by introducing a new variable name "x" to take the place of J^¢, and the region that represents this fresh featured x is circumscribed in a more conventional symmetry of form and placement. Just as we once gave the name "J" to the proposition u.v, we now give the name "!j!" to its thematization ((x, u v)). Already, again, at this culminating stage of reflection, we begin to think of the newly named proposition as a distinctive individual, a particular function !j! : <|u, v, x|> -> B. From now on, the terms "thematic extension" and "thematization" will be used to describe both the process and degree of explication that progresses through this series of pictures, both the operation of increasingly explicit symbolization and the dimension of variation that is swept out by it. To speak of this change in general, that takes us in our current example from J to !j!, I introduce a class of operators symbolized by the Greek letter theta, writing !j! = theta(J) in the present instance. The operator theta, in the present situation bearing the type theta : [u, v] -> [u, v, x], provides us with a convenient way of recapitulating and summarizing the complete cycle of thematic developments. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D30 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Thematization: Venn Diagrams (concl.) Figure 21 shows how the thematic extension operator theta acts on two further examples, the disjunction ((u)(v)) and the equality ((u, v)). Referring to the disjunction as f<u, v> and the equality as g<u, v>, I write the thematic extensions as !f! = theta(f) and !g! = theta(g). f g o-------------------------------o o-------------------------------o | | |```````````````````````````````| | o-----o o-----o | |```````o-----o```o-----o```````| | /```````\ /```````\ | |``````/ \`/ \``````| | /`````````o`````````\ | |`````/ o \`````| | /`````````/`\`````````\ | |````/ /`\ \````| | /`````````/```\`````````\ | |```/ /```\ \```| | o`````````o`````o```````` o | |``o o`````o o``| | |`````````|`````|`````````| | |``| |`````| |``| | |``` u ```|`````|``` v ```| | |``| u |`````| v |``| | |`````````|`````|`````````| | |``| |`````| |``| | o`````````o`````o`````````o | |``o o`````o o``| | \`````````\```/`````````/ | |```\ \```/ /```| | \`````````\`/`````````/ | |````\ \`/ /````| | \`````````o`````````/ | |`````\ o /`````| | \```````/ \```````/ | |``````\ /`\ /``````| | o-----o o-----o | |```````o-----o```o-----o```````| | | |```````````````````````````````| o-------------------------------o o-------------------------------o ((u)(v)) ((u , v)) | | | | theta theta | | | | v v !f! !g! o-------------------------------o o-------------------------------o |```````````````````````````````| | | |````````````o-----o````````````| | o-----o | |```````````/ \```````````| | /```````\ | |``````````/ \``````````| | /`````````\ | |`````````/ \`````````| | /```````````\ | |````````/ \````````| | /`````````````\ | |```````o f o```````| | o`````` g ``````o | |```````| |```````| | |```````````````| | |```````| |```````| | |```````````````| | |```````| |```````| | |```````````````| | |```````o-----o o-----o```````| | o-----o```o-----o | |``````/ \`````\ /`````/ \``````| | /`\ \`/ /`\ | |`````/ \`````o`````/ \`````| | /```\ o /```\ | |````/ \```/`\```/ \````| | /`````\ /`\ /`````\ | |```/ \`/```\`/ \```| | /```````\ /```\ /```````\ | |``o o-----o o``| | o`````````o-----o`````````o | |``| | | |``| | |`````````| |`````````| | |``| u | | v |``| | |``` u ```| |``` v ```| | |``| | | |``| | |`````````| |`````````| | |``o o o o``| | o`````````o o`````````o | |```\ \ / /```| | \`````````\ /`````````/ | |````\ \ / /````| | \`````````\ /`````````/ | |`````\ o /`````| | \`````````o`````````/ | |``````\ /`\ /``````| | \```````/ \```````/ | |```````o-----o```o-----o```````| | o-----o o-----o | |```````````````````````````````| | | o-------------------------------o o-------------------------------o ((f , ((u)(v)) )) ((g , ((u , v)) )) Figure 21. Thematization of Disjunction and Equality o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D31 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Thematization: Truth Tables | That which distorts honest shapes or which creates unearthly | beings or places or contingencies is a nuisance and a revolt. | | Walt Whitman, 'Leaves of Grass', [Whi, 19] Tables 22 through 25 outline a method for computing the thematic extensions of propositions in terms of their coordinate values. A preliminary step, as illustrated in Table 22, is to write out the truth table representations of the propositional forms whose thematic extensions one wants to compute, in the present instance, the functions f<u, v> = ((u)(v)) and g<u, v> = ((u, v)). Table 22. Disjunction f and Equality g o-------------------o-------------------o | u v | f g | o-------------------o-------------------o | | | | 0 0 | 0 1 | | | | | 0 1 | 1 0 | | | | | 1 0 | 1 0 | | | | | 1 1 | 1 1 | | | | o-------------------o-------------------o Next, each propositional form is individually represented in the fashion shown in Tables 23-i and 23-ii, using "f" and "g" as function names and creating new variables x and y to hold the associated functional values. This pair of Tables outlines the first stage in the transition from the 2-dimensional universes of f and g to the 3-dimensional universes of theta(f) and theta(g). The top halves of the Tables replicate the truth table patterns for f and g in the form f : [u, v] -> [x] and g : [u, v] -> [y]. The bottom halves of the tables print the negatives of these pictures, as it were, and paste the truth tables for (f) and (g) under the copies for f and g. At this stage, the columns for theta(f) and theta(g) are appended almost as afterthoughts, amounting to indicator functions for the sets of ordered triples that make up the functions f and g. Tables 23-i and 23-ii. Thematics of Disjunction and Equality (1) o-----------------o-----------o o-----------------o-----------o | u v f | x !f! | | u v g | y !g! | o-----------------o-----------o o-----------------o-----------o | | | | | | | 0 0 --> | 0 1 | | 0 0 --> | 1 1 | | | | | | | | 0 1 --> | 1 1 | | 0 1 --> | 0 1 | | | | | | | | 1 0 --> | 1 1 | | 1 0 --> | 0 1 | | | | | | | | 1 1 --> | 1 1 | | 1 1 --> | 1 1 | | | | | | | o-----------------o-----------o o-----------------o-----------o | | | | | | | 0 0 | 1 0 | | 0 0 | 0 0 | | | | | | | | 0 1 | 0 0 | | 0 1 | 1 0 | | | | | | | | 1 0 | 0 0 | | 1 0 | 1 0 | | | | | | | | 1 1 | 0 0 | | 1 1 | 0 0 | | | | | | | o-----------------o-----------o o-----------------o-----------o All the data is now in place to give the truth tables for theta(f) and theta(g). In the remaining steps all we do is to permute the rows and change the roles of x and y from dependent to independent variables. In Tables 24-i and 24-ii the rows are arranged in such a way as to put the 3-tuples <u, v, x> and <u, v, y> in binary numerical order, suitable for viewing as the arguments of the maps theta(f) = !f! : [u, v, x] -> B and theta(g) = !g! : [u, v, y]->B. Moreover, the structure of the tables is altered slightly, allowing the now vestigial functions f and g to be passed over without further attention and shifting the heavy vertical bars a notch to the right. In effect, this clinches the fact that the thematic variables x := f^¢ and y := g^¢ are now to be regarded as independent variables. Tables 24-i and 24-ii. Thematics of Disjunction and Equality (2) o-----------------o-----------o o-----------------o-----------o | u v f | x !f! | | u v g | y !g! | o-----------------o-----------o o-----------------o-----------o | | | | | | | 0 0 --> | 0 1 | | 0 0 | 0 0 | | | | | | | | 0 0 | 1 0 | | 0 0 --> | 1 1 | | | | | | | | 0 1 | 0 0 | | 0 1 --> | 0 1 | | | | | | | | 0 1 --> | 1 1 | | 0 1 | 1 0 | | | | | | | o-----------------o-----------o o-----------------o-----------o | | | | | | | 1 0 | 0 0 | | 1 0 --> | 0 1 | | | | | | | | 1 0 --> | 1 1 | | 1 0 | 1 0 | | | | | | | | 1 1 | 0 0 | | 1 1 | 0 0 | | | | | | | | 1 1 --> | 1 1 | | 1 1 --> | 1 1 | | | | | | | o-----------------o-----------o o-----------------o-----------o An optional reshuffling of the rows brings additional features of the thematic extensions to light. Leaving the columns in place for the sake of comparison, Tables 25-i and 25-ii sorts the rows in a different order, in effect treating x and y as the primary variables in their respective 3-tuples. Regarding the thematic extensions in the form !f! : [x, u, v] -> B and !g! : [y, u, v] -> B makes it easier to see in this tabular setting a property that was graphically obvious in the venn diagrams above. Specifically, when the thematic variable F^¢ is true then theta(F) exhibits the pattern of the original F, and when F^¢ is false then theta(F) exhibits the pattern of its negation (F). Tables 25-i and 25-ii. Thematics of Disjunction and Equality (3) o-----------------o-----------o o-----------------o-----------o | u v f | x !f! | | u v g | y !g! | o-----------------o-----------o o-----------------o-----------o | | | | | | | 0 0 --> | 0 1 | | 0 0 | 0 0 | | | | | | | | 0 1 | 0 0 | | 0 1 --> | 0 1 | | | | | | | | 1 0 | 0 0 | | 1 0 --> | 0 1 | | | | | | | | 1 1 | 0 0 | | 1 1 | 0 0 | | | | | | | o-----------------o-----------o o-----------------o-----------o | | | | | | | 0 0 | 1 0 | | 0 0 --> | 1 1 | | | | | | | | 0 1 --> | 1 1 | | 0 1 | 1 0 | | | | | | | | 1 0 --> | 1 1 | | 1 0 | 1 0 | | | | | | | | 1 1 --> | 1 1 | | 1 1 --> | 1 1 | | | | | | | o-----------------o-----------o o-----------------o-----------o Finally, Tables 26-i and 26-ii compare the tacit extensions !e! : [u, v] -> [u, v, x] and !e! : [u, v]->[u, v, y] with the thematic extensions of the same types, as applied to the propositions f and g, respectively. Tables 26-i and 26-ii. Tacit Extension and Thematization o-----------------o-----------o o-----------------o-----------o | u v x | !e!f !f! | | u v y | !e!g !g! | o-----------------o-----------o o-----------------o-----------o | | | | | | | 0 0 0 | 0 1 | | 0 0 0 | 1 0 | | | | | | | | 0 0 1 | 0 0 | | 0 0 1 | 1 1 | | | | | | | | 0 1 0 | 1 0 | | 0 1 0 | 0 1 | | | | | | | | 0 1 1 | 1 1 | | 0 1 1 | 0 0 | | | | | | | o-----------------o-----------o o-----------------o-----------o | | | | | | | 1 0 0 | 1 0 | | 1 0 0 | 0 1 | | | | | | | | 1 0 1 | 1 1 | | 1 0 1 | 0 0 | | | | | | | | 1 1 0 | 1 0 | | 1 1 0 | 1 0 | | | | | | | | 1 1 1 | 1 1 | | 1 1 1 | 1 1 | | | | | | | o-----------------o-----------o o-----------------o-----------o o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D32 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Thematization: Truth Tables (cont.) Table 27 summarizes the thematic extensions of all propositions on two variables. Column 4 lists the equations of form (( f^¢ , f^¢ <u, v> )) and Column 5 simplifies these equations into the form of algebraic expressions. (As always, "+" refers to exclusive disjunction, and "f" should be read as "[f_i]^¢" in the body of the Table.) Table 27. Thematization of Bivariate Propositions o---------o---------o----------o--------------------o--------------------o | u : 1 1 0 0 | f | theta (f) | theta (f) | | v : 1 0 1 0 | | | | o---------o---------o----------o--------------------o--------------------o | | | | | | | f_0 | 0 0 0 0 | () | (( f , () )) | f + 1 | | | | | | | | f_1 | 0 0 0 1 | (u)(v) | (( f , (u)(v) )) | f + u + v + uv | | | | | | | | f_2 | 0 0 1 0 | (u) v | (( f , (u) v )) | f + v + uv + 1 | | | | | | | | f_3 | 0 0 1 1 | (u) | (( f , (u) )) | f + u | | | | | | | | f_4 | 0 1 0 0 | u (v) | (( f , u (v) )) | f + u + uv + 1 | | | | | | | | f_5 | 0 1 0 1 | (v) | (( f , (v) )) | f + v | | | | | | | | f_6 | 0 1 1 0 | (u, v) | (( f , (u, v) )) | f + u + v + 1 | | | | | | | | f_7 | 0 1 1 1 | (u v) | (( f , (u v) )) | f + uv | | | | | | | o---------o---------o----------o--------------------o--------------------o | | | | | | | f_8 | 1 0 0 0 | u v | (( f , u v )) | f + uv + 1 | | | | | | | | f_9 | 1 0 0 1 | ((u, v)) | (( f , ((u, v)) )) | f + u + v | | | | | | | | f_10 | 1 0 1 0 | v | (( f , v )) | f + v + 1 | | | | | | | | f_11 | 1 0 1 1 | (u (v)) | (( f , (u (v)) )) | f + u + uv | | | | | | | | f_12 | 1 1 0 0 | u | (( f , u )) | f + u + 1 | | | | | | | | f_13 | 1 1 0 1 | ((u) v) | (( f , ((u) v) )) | f + v + uv | | | | | | | | f_14 | 1 1 1 0 | ((u)(v)) | (( f , ((u)(v)) )) | f + u + v + uv + 1 | | | | | | | | f_15 | 1 1 1 1 | (()) | (( f , (()) )) | f | | | | | | | o---------o---------o----------o--------------------o--------------------o o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D33 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Thematization: Truth Tables (concl.) In order to show what all of the thematic extensions from two dimensions to three dimensions look like in terms of coordinates, Tables 28 and 29 present ordinary truth tables for the functions f_i : B^2 -> B and for the corresponding thematizations theta(f_i) = !f!_i : B^3 -> B. Table 28. Propositions on Two Variables o-------o-----o----------------------------------------------------------------o | u v | | f f f f f f f f f f f f f f f f | | | | 00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 | o-------o-----o----------------------------------------------------------------o | | | | | 0 0 | | 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 | | | | | | 0 1 | | 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 | | | | | | 1 0 | | 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 | | | | | | 1 1 | | 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 | | | | | o-------o-----o----------------------------------------------------------------o Table 29. Thematic Extensions of Bivariate Propositions o-------o-----o----------------------------------------------------------------o | u v | f^¢ |!f! !f! !f! !f! !f! !f! !f! !f! !f! !f! !f! !f! !f! !f! !f! !f! | | | | 00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 | o-------o-----o----------------------------------------------------------------o | | | | | 0 0 | 0 | 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 | | | | | | 0 0 | 1 | 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 | | | | | | 0 1 | 0 | 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 | | | | | | 0 1 | 1 | 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 | | | | | | 1 0 | 0 | 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 | | | | | | 1 0 | 1 | 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 | | | | | | 1 1 | 0 | 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 | | | | | | 1 1 | 1 | 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 | | | | | o-------o-----o----------------------------------------------------------------o o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D34 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | If only the word 'artificial' were associated with | the idea of 'art', or expert skill gained through | voluntary apprenticeship (instead of suggesting | the factitious and unreal), we might say that | 'logical' refers to artificial thought. | | John Dewey, 'How We Think', [Dew, 56-57] Propositional Transformations In this Subdivision I develop a comprehensive set of concepts for dealing with transformations between universes of discourse. In this most general context the source and the target universes of a transformation are allowed to be distinct, but may also be one and the same. When these concepts are applied to dynamic systems one focuses on the important special cases of transformations that map a universe into itself, and transformations of this shape may be interpreted as the state transitions of a discrete dynamical process, as these take place among the myriad ways that a universe of discourse might change, and by that change turn into itself. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D35 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Alias and Alibi Transformations There are customarily two modes of understanding a transformation, at least, when we try to interpret its relevance to something in reality. A transformation always refers to a changing prospect, to say it in a unified but equivocal way, but this can be taken to mean either a subjective change in the interpreting observer's point of view or an objective change in the systematic subject of discussion. In practice these variant uses of the transformation concept are distinguished in the following terms: 1. A "perspectival" or "alias" transformation refers to a shift in perspective or a change in language that takes place in the observer's frame of reference. 2. A "transitional" or "alibi" transformation refers to a change of position or an alteration of state that occurs in the object system as it falls under study. (For a recent discussion of the "alias vs. alibi" issue, as it relates to linear transformations in vector spaces and to other issues of an algebraic nature, see [MaB, 256, 582-4].) Naturally, when we are concerned with the dynamical properties of a system, the transitional aspect of transformation is the factor that comes to the fore, and this involves us in contemplating all of the ways of changing a universe into itself while remaining under the rule of established dynamical laws. In the prospective application to dynamic systems, and to neural networks viewed in this light, our interest lies chiefly with the transformations of a state space into itself that constitute the state transitions of a discrete dynamic process. Nevertheless, many important properties of these transformations, and some constructions that we need to see most clearly, are independent of the transitional interpretation and are likely to be confounded with irrelevant features if presented first and only in that association. In addition, and in partial contrast, intelligent systems are exactly that species of dynamic agents that have the capacity to have a point of view, and we cannot do justice to their peculiar properties without examining their ability to form and transform their own frames of reference in exposure to the elements of their own experience. In this setting, the perspectival aspect of transformation is the facet that shines most brightly, perhaps too often leaving us fascinated with mere glimmerings of its actual potential. It needs to be emphasized that nothing of the ordinary sort needs be moved in carrying out a transformation under the alias interpretation, that it may only involve a change in the forms of address, an amendment of the terms which are customed to approach and fashioned to describe the very same things in the very same world. But again, working within a discipline of realistic computation, we know how formidably complex and resource-consuming such transformations of perspective can be to implement in practice, much less to endow in the self-governed form of a nascently intelligent dynamical system. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D36 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Transformations of General Type | 'Es ist passiert', "it just sort of happened", people said there | when other people in other places thought heaven knows what had | occurred. It was a peculiar phrase, not known in this sense to | the Germans and with no equivalent in other languages, the very | breath of it transforming facts and the bludgeonings of fate | into something light as eiderdown, as thought itself. | | Robert Musil, 'The Man Without Qualities', [Mus, 34] Consider the situation illustrated in Figure 30, where the alphabets !U! = {u, v} and !X! = {x, y, z} are used to label basic features in two different logical universes, U% = [u, v] and X% = [x, y, z]. o-------------------------------------------------------o | U | | | | o-----------o o-----------o | | / \ / \ | | / o \ | | / / \ \ | | / / \ \ | | o o o o | | | | | | | | | u | | v | | | | | | | | | o o o o | | \ \ / / | | \ \ / / | | \ o / | | \ / \ / | | o-----------o o-----------o | | | | | o---------------------------o---------------------------o / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ o-------------------------o o-------------------------o o-------------------------o | U | | U | | U | | o---o o---o | | o---o o---o | | o---o o---o | | / \ / \ | | / \ / \ | | / \ / \ | | / o \ | | / o \ | | / o \ | | / / \ \ | | / / \ \ | | / / \ \ | | o o o o | | o o o o | | o o o o | | | u | | v | | | | u | | v | | | | u | | v | | | o o o o | | o o o o | | o o o o | | \ \ / / | | \ \ / / | | \ \ / / | | \ o / | | \ o / | | \ o / | | \ / \ / | | \ / \ / | | \ / \ / | | o---o o---o | | o---o o---o | | o---o o---o | | | | | | | o-------------------------o o-------------------------o o-------------------------o \ | \ / | / \ | \ / | / \ | \ / | / \ | \ / | / \ g | \ f / | h / \ | \ / | / \ | \ / | / \ | \ / | / \ | \ / | / \ o----------|-----------\-----/-----------|----------o / \ | X | \ / | | / \ | | \ / | | / \ | | o-----o-----o | | / \| | / \ | |/ \ | / \ | / |\ | / \ | /| | \ | / \ | / | | \ | / \ | / | | \ | o x o | / | | \ | | | | / | | \ | | | | / | | \ | | | | / | | \ | | | | / | | \ | | | | / | | \| | | |/ | | o--o--------o o--------o--o | | / \ \ / / \ | | / \ \ / / \ | | / \ o / \ | | / \ / \ / \ | | / \ / \ / \ | | o o--o-----o--o o | | | | | | | | | | | | | | | | | | | | | y | | z | | | | | | | | | | | | | | | o o o o | | \ \ / / | | \ \ / / | | \ o / | | \ / \ / | | \ / \ / | | o-----------o o-----------o | | | | | o---------------------------------------------------o \ / \ / \ / \ / \ / \ p , q / \ / \ / \ / \ / \ / \ / \ / o Figure 30. Generic Frame of a Logical Transformation Enter the picture, as we usually do, in the middle of things, with features like x, y, z that present themselves to be simple enough in their own right and that form a satisfactory, if a temporary, foundation to provide a basis for discussion. In this universe and on these terms we find expression for various propositions and questions of principal interest to ourselves, as indicated by the maps p, q : X -> B. Then we discover that the simple features {x, y, z} are really more complex than we thought at first, and it becomes useful to regard them as functions {f, g, h} of other features {u, v}, that we place in a preface to our original discourse, or suppose as topics of a preliminary universe of discourse U% = [u, v]. It may happen that these late-blooming but pre-ambling features are found to lie closer, in a sense that may be our job to determine, to the central nature of the situation of interest, in which case they earn our regard as being more fundamental, but these functions and features are only required to supply a critical stance on the universe of discourse or an alternate perspective on the nature of things in order to be preserved as useful. A particular transformation F : [u, v] -> [x, y, z] may be expressed by a system of equations, as shown below. Here, F is defined by its component maps F = <F_1, F_2, F_3> = <f, g, h>, where each component map in {f, g, h} is a proposition of type B^n -> B^1. o-------------------------------------------------o | | | x = f<u, v> | | | | y = g<u, v> | | | | z = h<u, v> | | | o-------------------------------------------------o Regarded as a logical statement, this system of equations expresses a relation between a collection of freely chosen propositions {f, g, h} in one universe of discourse and the special collection of simple propositions {x, y, z} on which are founded another universe of discourse. Growing familiarity with a particular transformation of discourse, and the desire to achieve a ready understanding of its implications, requires that we be able to convert this information about generals and simples into information about all the main subtypes of propositions, including the linear and singular propositions. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D37 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | 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. | | C.S. Peirce, "The Maxim of Pragmatism", CP 5.438 Analytic Expansions: Operators and Functors Given the barest idea of a logical transformation, as suggested by the sketch in Figure 30, and having conceptualized the universe of discourse, with all of its points and propositions, as a beginning object of discussion, we are ready to enter the next phase of our investigation. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D38 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Operators on Propositions and Transformations The next step is naturally inclined toward objects of the next higher order, namely, with operators that take in argument lists of logical transformations and that give back specified types of logical transformations as their results. For our present aims, we do not need to consider the most general class of such operators, nor any one of them for its own sake. Rather, we are interested in the special sorts of operators that arise in the study and analysis of logical transformations. Figuratively speaking, these operators serve as instruments for the live tomography (and hopefully not the vivisection) of the forms of change under view. Beyond that, they open up ways to implement the changes of view that we need to grasp all the variations on a transformational theme, or to appreciate enough of its significant features to "get the drift" of the change occurring, to form a passing acquaintance or a synthetic comprehension of its general character and disposition. The simplest type of operator is one that takes a single transformation as an argument and returns a single transformation as a result, and most of the operators that I will explicitly consider here are of this kind. Figure 31 illustrates the typical situation. o---------------------------------------o | | | | | U% F X% | | o------------------>o | | | | | | | | | | | | | | | | | | !W! | | !W! | | | | | | | | | | | | | | v v | | o------------------>o | | !W!U% !W!F !W!X% | | | | | o---------------------------------------o Figure 31. Operator Diagram (1) In this Figure, "!W!" serves as a generic name for an operator, in this case one that takes a logical transformation F of type (U% -> X%) into a logical transformation !W!F of the type (!W!U% -> !W!X%). Thus, the operator !W! must be viewed as making assignments for both families of objects that we have previously considered, both for universes of discourse like U% and X% and for logical transformations like F. NB. Strictly speaking, an operator like !W! works between two whole categories of universes and transformations, which we call the "source" and the "target" categories of !W!. Given this setting, !W! specifies for each universe U% in its source category a definite universe !W!U% in its target category, and to each transformation F in its source category it assigns a unique transformation !W!F in its target category. Naturally, this only works if !W! takes the source U% and the target X% of the map F over to the source !W!U% and the target !W!X% of the map !W!F. With luck or care enough, we can avoid ever having to put anything like that in words again, letting diagrams do the work. In the situations of present concern we are usually focused on a single transformation F, and thus we can take it for granted that the assignment of universes under !W! is defined appropriately at the source and the target ends of F. It is not always the case, though, that we need to use the particular names (like "!W!U%" and "!W!X%") that !W! assigns by default to its operative image universes. In most contexts we will usually have a prior acquaintance with these universes under other names, and it is only necessary that we can tell from the information associated with an operator !W! what universes they are. In Figure 31 the maps F and !W!F are displayed horizontally, the way that one normally orients functional arrows in a written text, and !W! rolls the map F downward into the images that are associated with !W!F. In Figure 32 the same information is redrawn so that the maps F and !W!F flow down the page, and !W! unfurls the map F rightward into domains that are the eminent purview of !W!F. o---------------------------------------o | | | | | U% !W! !W!U% | | o------------------>o | | | | | | | | | | | | | | | | | | F | | !W!F | | | | | | | | | | | | | | v v | | o------------------>o | | X% !W! !W!X% | | | | | o---------------------------------------o Figure 32. Operator Diagram (2) The latter arrangement, as it appears in Figure 32, is more congruent with the thinking about operators that we shall be doing in the rest of this discussion, since all logical transformations from here on out will be pictured vertically, after the fashion of Figure 30. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D39 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Differential Analysis of Propositions and Transformations | The resultant metaphysical problem now is this: | | 'Does the man go round the squirrel or not?' | | William James, 'Pragmatism', [Jam, 43] The approach to the differential analysis of logical propositions and transformations of discourse that will be pursued here is carried out in terms of particular operators !W! that act on propositions F or on transformations F to yield the corresponding operator maps !W!F. The operator results then become the subject of a series of further stages of analysis, which take them apart into their propositional components, rendering them as a set of purely logical constituents. After this is done, all the parts are then re-integrated to reconstruct the original object in the light of a more complete understanding, at least in ways that enable one to appreciate certain aspects of it with fresh insight. NB. Remark on Strategy. At this point I run into a set of conceptual difficulties that force me to make a strategic choice in how I proceed. Part of the problem can be remedied by extending my discussion of tacit extensions to the transformational context. But the troubles that remain are much more obstinate and lead me to try two different types of solution. The approach that I develop first makes use of a variant type of extension operator, the "trope extension", to be defined below. This method is more conservative and requires less preparation, but has features which make it seem unsatisfactory in the long run. A more radical approach, but one with a better hope of long term success, makes use of the notion of "contingency spaces". These are an even more generous type of extended universe than the kind I currently use, but are defined subject to certain internal constraints. The extra work needed to set up this method forces me to put it off to a later stage. However, as a compromise, and to prepare the ground for the next pass, I call attention to the various conceptual difficulties as they arise along the way and try to give an honest estimate of how well my first approach deals with them. I now describe in general terms the particular operators that are instrumental to this form of analysis. The main series of operators all have the form !W! : (U% -> X%) -> (EU% -> EX%). If we assume that the source universe U% and the target universe X% have finite dimensions n and k, respectively, then each operator !W! is encompassed by the same abstract type: !W! : ([B^n] -> [B^k]) -> ([B^n x D^n] -> [B^k x D^k]). Since the range features of the operator result !W!F : [B^n x D^n] -> [B^k x D^k] can be sorted out by their ordinary versus their differential qualities and the component maps can be examined independently, the complete operator !W! can be separated accordingly into two components, in the form !W! = <!e!, W>. Given a fixed context of source and target universes of discourse, !e! is always the same type of operator, a multiple component elaboration of the tacit extension operators that were articulated earlier. In this context !e! has the shape: Concrete type. !e! : ( U% -> X% ) -> ( EU% -> X% ) Abstract type. !e! : ([B^n] -> [B^k]) -> ([B^n x D^n] -> [B^k]) On the other hand, the operator W is specific to each !W!. In this context W always has the form: Concrete type. W : ( U% -> X% ) -> ( EU% -> dX% ) Abstract type. W : ([B^n] -> [B^k]) -> ([B^n x D^n] -> [D^k]) In the types just assigned to !e! and W, and implicitly to their results !e!F and WF, I have listed the most restrictive ranges defined for them, rather than the more expansive target spaces that subsume these ranges. When there is need to recognize both, we may use type indications like the following: !e!F : (EU% -> X% c EX%) ~=~ ([B^n x D^n] -> [B^k] c [B^k x D^k]) WF : (EU% -> dX% c EX%) ~=~ ([B^n x D^n] -> [D^k] c [B^k x D^k]) Hopefully, though, a general appreciation of these subsumptions will prevent us from having to make such declarations more often than absolutely necessary. In giving names to these operators I am attempting to preserve as much of the traditional nomenclature and as many of the classical associations as possible. The chief difficulty in doing this is occasioned by the distinction between the "sans serif" operators !W! and their "serified" components W, which forces me to find two distinct but parallel sets of terminology. Here is the plan that I have settled on. First, the component operators W are named by analogy with the corresponding operators in the classical difference calculus. Next, the complete operators !W! = <!e!, W> are assigned their titles according to their roles in a geometric or trigonometric allegory, if only to ensure that the tangent functor, that belongs to this family and whose exposition I am still working toward, comes out fit with its customary name. Finally, the operator results !W!F and WF can be fixed in this frame of reference by tethering the operative adjective for !W! or W to the anchoring epithet "map", in conformity with an already standard practice. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D40 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o The Secant Operator: $E$ | Mr. Peirce, after pointing out that our beliefs are really | rules for action, said that, to develop a thought's meaning, | we need only determine what conduct it is fitted to produce: | that conduct is for us its sole significance. | | William James, 'Pragmatism', [Jam, 46] Figures 33-i and 33-ii depict two stages in the form of analysis that will be applied to transformations throughout the remainder of this study. From now on our interest is staked on an operator denoted "$E$", which receives the principal investment of analytic attention, and on the constituent parts of $E$, which derive their shares of significance as developed by the analysis. In the sequel, I refer to $E$ as the "secant operator", taking it for granted that a context has been chosen that defines its type. The secant operator has the component description $E$ = <!e!, E>, and its active ingredient E is known as the "enlargement operator". (Here, I have named E after the literal ancestor of the shift operator in the calculus of finite differences, defined so that Ef(x) = f(x+1) for any suitable function f, though, of course the logical analogue that we take up here must have a rather different definition.) U% $E$ $E$U% $E$U% $E$U% o------------------>o============o============o | | | | | | | | | | | | | | | | F | | $E$F = | $d$^0.F + | $r$^0.F | | | | | | | | | | | | v v v v o------------------>o============o============o X% $E$ $E$X% $E$X% $E$X% Figure 33-i. Analytic Diagram (1) U% $E$ $E$U% $E$U% $E$U% $E$U% o------------------>o============o============o============o | | | | | | | | | | | | | | | | | | | | F | | $E$F = | $d$^0.F + | $d$^1.F + | $r$^1.F | | | | | | | | | | | | | | | v v v v v o------------------>o============o============o============o X% $E$ $E$X% $E$X% $E$X% $E$X% Figure 33-ii. Analytic Diagram (2) In its action on universes $E$ yields the same result as E, a fact that can be expressed in equational form by writing $E$U% = EU% for any universe U%. Notice that the extended universes across the top and bottom of the diagram are indicated to be strictly identical, rather than requiring a corresponding decomposition for them. In a certain sense, the functional parts of $E$F are partitioned into separate contexts that have to be re-integrated again, but the best image to use is that of making transparent copies of each universe and then overlapping their functional contents once more at the conclusion of the analysis, as suggested by the graphic conventions that are used at the top of Figure 30. Acting on a transformation F from universe U% to universe X%, the operator $E$ determines a transformation $E$F from $E$U% to $E$X%. The map $E$F forms the main body of evidence to be investigated in performing a differential analysis of F. Because we shall frequently be focusing on small pieces of this map for considerable lengths of time, and consequently lose sight of the "big picture", it is critically important to emphasize that the map $E$F is a transformation that determines a relation from one extended universe into another. This means that we should not be satisfied with our understanding of a transformation F until we can lay out the full "parts diagram" of $E$F along the lines of the generic frame in Figure 30. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D41 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o The Secant Operator: $E$ (concl.) If one is working within the confines of propositional calculus, it is possible to give an elementary definition of $E$F by means of a system of propositional equations, as will now be described. Given a transformation: F = <F_1, ..., F_k> : B^n -> B^k of concrete type: F : [u_1, ..., u_n] -> [x_1, ..., x_k] the transformation: $E$F = <F_1, ..., F_k, EF_1, ..., EF_k> : B^n x D^n -> B^k x D^k of concrete type: $E$F : [u_1, ..., u_n, du_1, ..., du_n] -> [x_1, ..., x_k, dx_1, ..., dx_k] is defined by means of the following system of logical equations: o--------------------------------------------------------------------------------------o | | | x_1 = !e!F_1 <u_1, ..., u_n, du_1, ..., du_n> = F_1 <u_1, ..., u_n> | | | | ... | | | | x_k = !e!F_k <u_1, ..., u_n, du_1, ..., du_n> = F_k <u_1, ..., u_n> | | | | | | dx_1 = EF_1 <u_1, ..., u_n, du_1, ..., du_n> = F_1 <u_1 + du_1, ..., u_n + du_n> | | | | ... | | | | dx_k = EF_k <u_1, ..., u_n, du_1, ..., du_n> = F_k <u_1 + du_1, ..., u_n + du_n> | | | o--------------------------------------------------------------------------------------o It is important to note that this system of equations can be read as a conjunction of equational propositions, in effect, as a single proposition in the universe of discourse that is generated by all of the named variables. Specifically, this is the universe of discourse over 2(n + k) variables that is denoted by: E[!U! |_| !X!] = [u_1, ..., u_n, x_1, ..., x_k, du_1, ..., du_n, dx_1, ..., dx_k]. In this light, it should be clear that the system of equations defining $E$F embodies, in a higher rank and in a differentially extended version, an analogy with the process of thematization that was treated earlier for propositions of the type F : B^n -> B. The entire collection of constraints that is represented in the above system of equations may be abbreviated by writing $E$F = <!e!F, EF>, for any map F. This is tantamount to regarding $E$ as a complex operator, $E$ = <!e!, E>, with a form of application that distributes each component of the operator to work on each component of the operand: $E$F = <!e!, E> F = <!e!F, EF> = <!e!F_1, ..., !e!F_k, EF_1, ..., EF_k>. Quite a lot of "thematic infrastructure" or interpretive information is being swept under the rug in the use of such abbreviations. When confusion arises about the meaning of such constructions, one always has recourse to the defining system of equations, in its totality a purely propositional expression. This means that the angle brackets, which were used in this context to build an image of multi-component transformations, should not be expected to determine a well-defined product in themselves, but only to serve as reminders of the prior thematic decisions (choices of variable names, etc.) that have to be made in order to determine one. Accordingly, the angle bracket notation < , > can be regarded as a kind of "thematic frame", an interpretive storage device that preserves the proper associations of concrete logical features between the extended universes at the source and target of $E$F. The generic notations $d$^0.F, $d$^1.F, ..., $d$^m.F in Figure 33 refer to the increasing orders of differentials that are extracted in the course of analyzing F. When the analysis is halted at a partial stage of development, notations like $r$^0.F, $r$^1.F, ..., $r$^m.F may be used to summarize the contributions to $E$F that remain to be analyzed. The Figure illustrates a convention that renders the remainder term $r$^m.F, in effect, the sum of all differentials of order strictly greater than m. I next discuss the set of operators that figure into this form of analysis, describing their effects on transformations. In simplified or specialized contexts these operators tend to take on a variety of different names and notations, some of whose number I will introduce along the way. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D42 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o The Radius Operator: $e$ | And the tangible fact at the root of all our thought-distinctions, | however subtle, is that there is no one of them so fine as to | consist in anything but a possible difference of practice. | | William James, 'Pragmatism', [Jam, 46] The operator identified as $d$^0 in the analytic diagram (Figure 33) has the sole purpose of creating a proxy for F in the appropriately extended context. Construed in terms of its broadest components, $d$^0 is equivalent to the doubly tacit extension operator <!e!, !e!>, in recognition of which let us redub it as "$e$". Pursuing a geometric analogy, we may refer to $e$ = <!e!, !e!> = $d$^0 as the "radius operator". The operation that is intended by all of these forms is defined by the equation: $e$F = <!e!, !e!> F = <!e!F, !e!F> = <!e!F_1, ..., !e!F_k, !e!F_1, ..., !e!F_k>, which is tantamount to the system of equations given below. o--------------------------------------------------------------------------------o | | | x_1 = !e!F_1 <u_1, ..., u_n, du_1, ..., du_n> = F_1 <u_1, ..., u_n> | | | | ... | | | | x_k = !e!F_k <u_1, ..., u_n, du_1, ..., du_n> = F_k <u_1, ..., u_n> | | | | | | dx_1 = !e!F_1 <u_1, ..., u_n, du_1, ..., du_n> = F_1 <u_1, ..., u_n> | | | | ... | | | | dx_k = !e!F_k <u_1, ..., u_n, du_1, ..., du_n> = F_k <u_1, ..., u_n> | | | o--------------------------------------------------------------------------------o o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D43 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o The Phantom of the Operators: !h! | I was wondering what the reason could be, | when I myself raised my head and everything | within me seemed drawn towards the Unseen, | 'which was playing the most perfect music'! | | Gaston Leroux, 'The Phantom of the Opera', [Ler, 81] I now describe an operator whose persistent but elusive action behind the scenes, whose slightly twisted and ambivalent character, and whose fugitive disposition, caught somewhere in flight between the arrantly negative and the positive but errant intent, has cost me some painstaking trouble to detect. In the end I shall place it among the other extensions and projections, as a shade among shadows, of muted tones and motley hue, that adumbrates its own thematic frame and paradoxically lights the way toward a whole new spectrum of values. Given a transformation F : [u_1, ..., u_n] -> [x_1, ..., x_k], we often need to make a separate treatment of a related family of transformations of the form F* : [u_1, ..., u_n, du_1, ..., du_n] -> [dx_1, ..., dx_k]. The operator !h! (Greek eta) is introduced to deal with the simplest one of these maps: !h!F : [u_1, ..., u_n, du_1, ..., du_n] -> [dx_1, ..., dx_k] which is defined by the equations: o--------------------------------------------------------------------------------o | | | dx_1 = !e!F_1 <u_1, ..., u_n, du_1, ..., du_n> = F_1 <u_1, ..., u_n> | | | | ... | | | | dx_k = !e!F_k <u_1, ..., u_n, du_1, ..., du_n> = F_k <u_1, ..., u_n> | | | o--------------------------------------------------------------------------------o In effect, the operator !h! is nothing but the stand-alone version of a procedure that is otherwise invoked subordinate to the work of the radius operator $e$. Operating independently, !h! achieves precisely the same results that the second !e! in <!e!, !e!> accomplishes by working within the context of its adjuvant thematic frame, "< , >". From this point on, because the use of !e! and !h! in this setting combines the aims of both the tacit and the thematic extensions, and because !h! reflects in regard to !e! little more than the application of a differential twist, a mere turn of phrase, I refer to !h! as the "trope extension" operator. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D44 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o The Chord Operator: $D$ | What difference would it practically make to any one if this | notion rather than that notion were true? If no practical | difference whatever can be traced, then the alternatives | mean practically the same thing, and all dispute is idle. | | William James, 'Pragmatism', [Jam, 45] Next I discuss an operator that is always immanent in this form of analysis, and remains implicitly present in the entire proceeding. It may appear once as a record: a relic or revenant that reprises the reminders of an earlier stage of development. Or it may appear always as a resource: a reserve or redoubt that caches in advance an echo of what remains to be played out, cleared up, and requited in full at a future stage. And all of this remains true whether or not we recall the key at any time, and whether or not the subtending theme is recited explicitly at any stage of play. This is the operator that is referred to as $r$^0 in the initial stage of analysis (Figure 33-i), and that is expanded as $d$^1 + $r$^1 in the subsequent step (Figure 33-ii). In congruence, but not quite harmony, with my allusions of analogy that are not quite geometry, I call this the "chord operator" and denote it $D$. In the more casual terms that are here introduced, $D$ is defined as the remainder of $E$ and $e$, and it assigns a due measure to each undertone of accord or discord that is struck between the note of enterprise $E$ and the bar of exigency $e$. The tension between these counterposed notions, in balance transient but regular in stridence, may be refracted along familiar lines, though never by any such fraction resolved. In this style we may write $D$ = <!e!, D>, calling D the "difference operator" and noting that it plays a role in this realm of mutable and diverse discourse that is analogous to the part taken by the discrete difference operator in the ordinary difference calculus. Finally, we should note that the chord $D$ is not one that need be lost at any stage of development. At the m^th stage of play it can always be reconstituted in the following form: o-------------------------------------------------o | | | $D$ = $E$ - $e$ | | | | = $r$^0 | | | | = $d$^1 + $r$^1 | | | | = Sum_(i = 1 to m) $d$^i + $r$^m | | | o-------------------------------------------------o o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D45 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o The Tangent Operator: $T$ | They take part in scenes of whose significance they have no inkling. | They are merely tangent to curves of history the beginnings and ends | and forms of which pass wholly beyond their ken. So we are tangent | to the wider life of things. | | William James, 'Pragmatism', [Jam, 300] The operator tagged as $d$^1 in the analytic diagram (Figure 33) is called the "tangent operator", and is usually denoted in this text as $d$ or $T$. Because it has the properties required to qualify as a functor, namely, preserving the identity element of the composition operation and the articulated form of every composure among transformations, it also earns the title of a "tangent functor". According to the custom adopted here, we dissect it as $T$ = $d$ = <!e!, d>, where d is the operator that yields the first order differential dF when applied to a transformation F, and whose name is legion. Figure 34 illustrates a stage of analysis where we ignore everything but the tangent functor $T$, and attend to it chiefly as it bears on the first order differential dF in the analytic expansion of F. In this situation, we often refer to the extended universes EU% and EX% under the equivalent designations $T$U% and $T$X%, respectively. The purpose of the tangent functor $T$ is to extract the tangent map $T$F at each point of U%, and the tangent map $T$F = <!e!, d> F tells us not only what the transformation F is doing at each point of the universe U% but also what F is doing to states in the neighborhood of that point, approximately, linearly, and relatively speaking. U% $T$ $T$U% $T$U% o------------------>o============o | | | | | | | | | | | | F | | $T$F = | <!e!, d> F | | | | | | | | | v v v o------------------>o============o X% $T$ $T$X% $T$X% Figure 34. Tangent Functor Diagram NB. There is one aspect of the preceding construction that remains especially problematic. Why did we define the operators W in {!h!, E, D, d, r} so that the ranges of their resulting maps all fall within the realms of differential quality, even fabricating a variant of the tacit extension operator to have that character? Clearly, not all of the operator maps WF have equally good reasons for placing their values in differential stocks. The only explanation I can devise at present is that, without doing this, I cannot justify the comparison and combination of their values in the various analytic steps. By default, only those values in the same functional component can be brought into algebraic modes of interaction. Up till now, the only mechanism provided for their broader association has been a purely logical one, their common placement in a target universe of discourse, but the task of converting this logical circumstance into algebraic forms of application has not yet been taken up. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D46 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Transformations of Type B^2 -> B^1 To study the effects of these analytic operators in the simplest possible situation, let us revert to a still more primitive case. Consider the singular proposition J<u, v> = uv, regarded either as the functional product of the maps u and v or as the logical conjunction of the features u and v, a map whose fiber of truth J^(-1)(1) picks out the single cell of that logical description in the universe of discourse U%. Thus J, or uv, may be treated as a pseudonym for the point whose coordinates are <1, 1> in U%. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D47 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Analytic Expansion of Conjunction | In her sufferings she read a great deal and discovered | that she had lost something, the possession of which | she had previously not been much aware of: a soul. | | What is that? It is easily defined negatively: | it is simply what curls up and hides when there | is any mention of algebraic series. | | Robert Musil, 'The Man Without Qualities', [Mus, 118] Figure 35 pictures the form of conjunction J : B^2 -> B as a transformation from the 2-dimensional universe [u, v] to the 1-dimensional universe [x]. This is a subtle but significant change of viewpoint on the proposition, attaching an arbitrary but concrete quality to its functional value. Using the language introduced earlier, we can express this change by saying that the proposition J : <|u, v|> -> B is being recast into the thematized role of a transformation J : [u, v] -> [x], where the new variable "x" takes the part of a thematic variable ¢(J). o---------------------------------------o | | | | | o---------o o---------o | | / \ / \ | | / o \ | | / /`\ \ | | / /```\ \ | | o o`````o o | | | |`````| | | | | u |`````| v | | | | |`````| | | | o o`````o o | | \ \```/ / | | \ \`/ / | | \ o / | | \ / \ / | | o---------o o---------o | | | | | o---------------------------------------o \ / \ / \ / \ J / \ / \ / \ / o--------------\---------/--------------o | \ / | | \ / | | o------@------o | | /```````````````\ | | /`````````````````\ | | /```````````````````\ | | /`````````````````````\ | | o```````````````````````o | | |```````````````````````| | | |`````````` x ``````````| | | |```````````````````````| | | o```````````````````````o | | \`````````````````````/ | | \```````````````````/ | | \`````````````````/ | | \```````````````/ | | o-------------o | | | | | o---------------------------------------o Figure 35. Conjunction as Transformation o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D48 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Tacit Extension of Conjunction | I teach straying from me, yet who can stray from me? | I follow you whoever you are from the present hour; | My words itch at your ears till you understand them. | | Walt Whitman, 'Leaves of Grass', [Whi, 83] Earlier I defined the tacit extension operators !e! : X% -> Y% as maps embedding each proposition of a given universe X% in a more generously given universe Y% containing X%. Of immediate interest are the tacit extensions !e! : U% -> EU%, that locate each proposition of U% in the enlarged context of EU%. In its application to the propositional conjunction J = u v in [u, v], the tacit extension operator !e! produces the proposition !e!J in EU% = [u, v, du, dv]. The extended proposition !e!J may be computed according to the scheme in Table 36, in effect, doing nothing more than conjoining a tautology of [du, dv] to J in U%. Table 36. Computation of !e!J o---------------------------------------------------------------------o | | | !e!J = J<u, v> | | | | = u v | | | | = u v (du)(dv) + u v (du) dv + u v du (dv) + u v du dv | | | o---------------------------------------------------------------------o | | | !e!J = u v (du)(dv) + | | u v (du) dv + | | u v du (dv) + | | u v du dv | | | o---------------------------------------------------------------------o The lower portion of the Table contains the dispositional features of !e!J arranged in such a way that the variety of ordinary features spreads across the rows and the variety of differential features runs through the columns. This organization serves to facilitate pattern matching in the remainder of our computations. Again, the tacit extension is usually so trivial a concern that we do not always bother to make an explicit note of it, taking it for granted that any function F that is being employed in a differential context is equivalent to !e!F, for a suitable !e!. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D49 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Tacit Extension of Conjunction (cont.) Figures 37-a through 37-d present several pictures of the proposition J and its tacit extension !e!J. Notice in these Figures how !e!J in EU% visibly extends J in U%, by annexing to the indicated cells of J all of the arcs that exit from or flow out of them. In effect, this extension attaches to these cells all of the dispositions that spring from them, in other words, it attributes to these cells all of the conceivable changes that are their issue. o---------------------------------------o | | | o | | /%\ | | /%%%\ | | /%%%%%\ | | o%%%%%%%o | | /%\%%%%%/%\ | | /%%%\%%%/%%%\ | | /%%%%%\%/%%%%%\ | | o%%%%%%%o%%%%%%%o | | / \%%%%%/%\%%%%%/ \ | | / \%%%/%%%\%%%/ \ | | / \%/%%%%%\%/ \ | | o o%%%%%%%o o | | / \ / \%%%%%/ \ / \ | | / \ / \%%%/ \ / \ | | / \ / \%/ \ / \ | | o o o o o | | |\ / \ / \ / \ /| | | | \ / \ / \ / \ / | | | | \ / \ / \ / \ / | | | | o o o o | | | | |\ / \ / \ /| | | | | | \ / \ / \ / | | | | | u | \ / \ / \ / | v | | | o---+---o o o---+---o | | | \ / \ / | | | | \ / \ / | | | | du \ / \ / dv | | | o-------o o-------o | | \ / | | \ / | | \ / | | o | | | o---------------------------------------o Figure 37-a. Tacit Extension of J (Areal) o-----------------------------o | | | o-----o o-----o | | / \ / \ | | / o \ | | / / \ \ | | o o o o | @ | du | | dv | | /| o o o o | / | \ \ / / | / | \ o / | / | \ / \ / | / | o-----o o-----o | / | | / o-----------------------------o / o----------------------------------------/----o o-----------------------------o | / | | | | @ | | o-----o o-----o | | | | / \ / \ | | o---------o o---------o | | / o \ | | / \ / \ | | / / \ \ | | / o \ | | o o o o | | / /`\ @------\-----------@ | du | | dv | | | / /```\ \ | | o o o o | | / /`````\ \ | | \ \ / / | | / /```````\ \ | | \ o / | | o o`````````o o | | \ / \ / | | | |````@````| | | | o-----o o-----o | | | |`````\```| | | | | | | |``````\``| | | o-----------------------------o | | u |```````\`| v | | | | |````````\| | | o-----------------------------o | | |`````````| | | | | | | |`````````|\ | | | o-----o o-----o | | o o`````````o \ o | | / \ / \ | | \ \```````/ \ / | | / o \ | | \ \`````/ \ / | | / / \ \ | | \ \```/ \ / | | o o o o | | \ @------\-/---------\---------------@ | du | | dv | | | \ o \ / | | o o o o | | \ / \ / | | \ \ / / | | o---------o o---------o \ | | \ o / | | \ | | \ / \ / | | \ | | o-----o o-----o | | \ | | | o----------------------------------------\----o o-----------------------------o \ \ o-----------------------------o \ |`````````````````````````````| \ |````` o-----o```o-----o``````| \ |`````/```````\`/```````\`````| \ |````/`````````o`````````\````| \ |```/`````````/`\`````````\```| \|``o`````````o```o`````````o``| @``|```du````|```|````dv```|``| |``o`````````o```o`````````o``| |```\`````````\`/`````````/```| |````\`````````o`````````/````| |`````\```````/`\```````/`````| |``````o-----o```o-----o``````| |`````````````````````````````| o-----------------------------o Figure 37-b. Tacit Extension of J (Bundle) o---------------------------------------------------------------------o | | | | | o-------------------o o-------------------o | | / \ / \ | | / o \ | | / / \ \ | | / / \ \ | | / / \ \ | | / / \ \ | | / / \ \ | | o o (du).(dv) o o | | | | -->-- | | | | | | \ / | | | | | dv .(du) | \ / | du .(dv) | | | | u <---------------@---------------> v | | | | | | | | | | | | | | | | | | | | | | | | o o | o o | | \ \ | / / | | \ \ | / / | | \ \ | / / | | \ \ | / / | | \ \|/ / | | \ | / | | \ /|\ / | | o-------------------o | o-------------------o | | | | | du . dv | | | | | V | | | o---------------------------------------------------------------------o Figure 37-c. Tacit Extension of J (Compact) o-----------------------------------------------------------o | | | (du).(dv) | | --->--- | | \ / | | \ / | | \ / | | u @ v | | /|\ | | / | \ | | / | \ | | / | \ | | / | \ | | (du) dv / | \ du (dv) | | / | \ | | / | \ | | / | \ | | / | \ | | v | v | | @ | @ | | u (v) | (u) v | | | | | | | | | | | | | | du . dv | | | | | | | | | | | | | | v | | @ | | | | (u).(v) | | | o-----------------------------------------------------------o Figure 37-d. Tacit Extension of J (Digraph) o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D50 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Tacit Extension of Conjunction (concl.) The computational scheme that was shown in Table 36 treated J as a proposition in U% and formed !e!J as a proposition in EU%. When J is regarded as a mapping J : U% -> X% then !e!J must be obtained as a mapping !e!J : EU% -> X%. By default, the tacit extension of the map J : [u, v] -> [x] is naturally taken to be a particular map, of the following form: !e!J : [u, v, du, dv] -> [x] c [x, dx] This is the map that looks like J when painted in the frame of the extended source universe and that takes the same thematic variable in the extended target universe as the one that J already employs. But the choice of a particular thematic variable, for example "x" for ¢(J), is a shade more arbitrary than the initial choice of variable names {u, v}. This means that the map I am calling the "trope extension", specifically: !h!J : [u, v, du, dv] -> [dx] c [x, dx] since it looks just the same as !e!J in the way that its fibers paint the source domain, belongs just as fully to the family of tacit extensions, generically considered. These considerations have the practical consequence that all of our computations and illustrations of !e!J perform the double duty of capturing an image of !h!J as well. In other words, we are saved the work of carrying out calculations and drawing figures for the trope extension !h!J, because the exercise would be identical to the work already done for !e!J. Since the computations given for !e!J are expressed solely in terms of the variables {u, v, du, dv}, these variables work equally well for finding !h!J. Furthermore, since each of the above Figures shows only how the level sets of !e!J partition the extended source universe EU% = [u, v, du, dv], all of them serve equally well as portraits of !h!J. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D51 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Enlargement Map of Conjunction | No one could have established the existence of any details | that might not just as well have existed in earlier times | too; but all the relations between things had shifted | slightly. Ideas that had once been of lean account | grew fat. | | Robert Musil, 'The Man Without Qualities', [Mus, 62] The enlargement map EJ is computed from the proposition J by making a particular class of formal substitutions for its variables, in this case "u + du" for "u" and "v + dv" for "v", and subsequently expanding the result in whatever way happens to be convenient for the end in view. Table 38 shows a typical scheme of computation, following a systematic method of exploiting boolean expansions over selected variables, and ultimately developing EJ over the cells of [u, v]. The critical step of this procedure uses the facts that (0, x) = 0 + x = x and (1, x) = 1 + x = (x) for any boolean variable x. Table 38. Computation of EJ (Method 1) o-------------------------------------------------------------------------------o | | | EJ = J<u + du, v + dv> | | | | = (u, du)(v, dv) | | | | = u v J<1 + du, 1 + dv> + | | | | u (v) J<1 + du, 0 + dv> + | | | | (u) v J<0 + du, 1 + dv> + | | | | (u)(v) J<0 + du, 0 + dv> | | | | = u v J<(du), (dv)> + | | | | u (v) J<(du), dv > + | | | | (u) v J< du , (dv)> + | | | | (u)(v) J< du , dv > | | | o-------------------------------------------------------------------------------o | | | EJ = u v (du)(dv) | | + u (v)(du) dv | | + (u) v du (dv) | | + (u)(v) du dv | | | o-------------------------------------------------------------------------------o Table 39 exhibits another method that happens to work quickly in this particular case, using distributive laws to multiply things out in an algebraic manner, arranging the notations of feature and fluxion according to a scale of simple character and degree. Proceeding this way leads through an intermediate step which, in chiming the changes of ordinary calculus, should take on a familiar ring. Consequential properties of exclusive disjunction then carry us on to the concluding line. Table 39. Computation of EJ (Method 2) o-------------------------------------------------------------------------------o | | | EJ = <u + du> <v + dv> | | | | = u v + u dv + v du + du dv | | | | EJ = u v (du)(dv) + u (v)(du) dv + (u) v du (dv) + (u)(v) du dv | | | o-------------------------------------------------------------------------------o o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D52 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Enlargement Map of Conjunction (concl.) Figures 40-a through 40-d present several views of the enlarged proposition EJ. o---------------------------------------o | | | o | | /%\ | | /%%%\ | | /%%%%%\ | | o%%%%%%%o | | / \%%%%%/ \ | | / \%%%/ \ | | / \%/ \ | | o o o | | /%\ / \ /%\ | | /%%%\ / \ /%%%\ | | /%%%%%\ / \ /%%%%%\ | | o%%%%%%%o o%%%%%%%o | | / \%%%%%/ \ / \%%%%%/ \ | | / \%%%/ \ / \%%%/ \ | | / \%/ \ / \%/ \ | | o o o o o | | |\ / \ /%\ / \ /| | | | \ / \ /%%%\ / \ / | | | | \ / \ /%%%%%\ / \ / | | | | o o%%%%%%%o o | | | | |\ / \%%%%%/ \ /| | | | | | \ / \%%%/ \ / | | | | | u | \ / \%/ \ / | v | | | o---+---o o o---+---o | | | \ / \ / | | | | \ / \ / | | | | du \ / \ / dv | | | o-------o o-------o | | \ / | | \ / | | \ / | | o | | | o---------------------------------------o Figure 40-a. Enlargement of J (Areal) o-----------------------------o | | | o-----o o-----o | | / \ / \ | | / o \ | | / /%\ \ | | o o%%%o o | @ | du |%%%| dv | | /| o o%%%o o | / | \ \%/ / | / | \ o / | / | \ / \ / | / | o-----o o-----o | / | | / o-----------------------------o / o----------------------------------------/----o o-----------------------------o | / | | | | @ | | o-----o o-----o | | | | /%%%%%%%\ / \ | | o---------o o---------o | | /%%%%%%%%%o \ | | / \ / \ | | /%%%%%%%%%/ \ \ | | / o \ | | o%%%%%%%%%o o o | | / /`\ @------\-----------@ |%% du %%%| | dv | | | / /```\ \ | | o%%%%%%%%%o o o | | / /`````\ \ | | \%%%%%%%%%\ / / | | / /```````\ \ | | \%%%%%%%%%o / | | o o`````````o o | | \%%%%%%%/ \ / | | | |````@````| | | | o-----o o-----o | | | |`````\```| | | | | | | |``````\``| | | o-----------------------------o | | u |```````\`| v | | | | |````````\| | | o-----------------------------o | | |`````````| | | | | | | |`````````|\ | | | o-----o o-----o | | o o`````````o \ o | | / \ /%%%%%%%\ | | \ \```````/ \ / | | / o%%%%%%%%%\ | | \ \`````/ \ / | | / / \%%%%%%%%%\ | | \ \```/ \ / | | o o o%%%%%%%%%o | | \ @------\-/---------\---------------@ | du | |%%% dv %%| | | \ o \ / | | o o o%%%%%%%%%o | | \ / \ / | | \ \ /%%%%%%%%%/ | | o---------o o---------o \ | | \ o%%%%%%%%%/ | | \ | | \ / \%%%%%%%/ | | \ | | o-----o o-----o | | \ | | | o----------------------------------------\----o o-----------------------------o \ \ o-----------------------------o \ |%%%%%%%%%%%%%%%%%%%%%%%%%%%%%| \ |%%%%%%o-----o%%%o-----o%%%%%%| \ |%%%%%/ \%/ \%%%%%| \ |%%%%/ o \%%%%| \ |%%%/ / \ \%%%| \|%%o o o o%%| @%%| du | | dv |%%| |%%o o o o%%| |%%%\ \ / /%%%| |%%%%\ o /%%%%| |%%%%%\ /%\ /%%%%%| |%%%%%%o-----o%%%o-----o%%%%%%| |%%%%%%%%%%%%%%%%%%%%%%%%%%%%%| o-----------------------------o Figure 40-b. Enlargement of J (Bundle) o---------------------------------------------------------------------o | | | | | o-------------------o o-------------------o | | / \ / \ | | / o \ | | / / \ \ | | / / \ \ | | / / \ \ | | / / \ \ | | / / \ \ | | o o (du).(dv) o o | | | | -->-- | | | | | | \ / | | | | | dv .(du) | \ / | du .(dv) | | | | u o---------------->@<----------------o v | | | | | ^ | | | | | | | | | | | | | | | | | | o o | o o | | \ \ | / / | | \ \ | / / | | \ \ | / / | | \ \ | / / | | \ \|/ / | | \ | / | | \ /|\ / | | o-------------------o | o-------------------o | | | | | du . dv | | | | | o | | | o---------------------------------------------------------------------o Figure 40-c. Enlargement of J (Compact) o-----------------------------------------------------------o | | | (du).(dv) | | --->--- | | \ / | | \ / | | \ / | | u @ v | | ^^^ | | / | \ | | / | \ | | / | \ | | / | \ | | (du) dv / | \ du (dv) | | / | \ | | / | \ | | / | \ | | / | \ | | / | \ | | @ | @ | | u (v) | (u) v | | | | | | | | | | | | | | du . dv | | | | | | | | | | | | | | | | | @ | | | | (u).(v) | | | o-----------------------------------------------------------o Figure 40-d. Enlargement of J (Digraph) An intuitive reading of the proposition EJ becomes available at this point, and may be useful. Recall that propositions in the extended universe EU% express the "dispositions" of system and the constraints that are placed on them. In other words, a differential proposition in EU% can be read as referring to various changes that a system might undergo in and from its various states. In particular, we can understand EJ as a statement that tells us what changes need to be made with regard to each state in the universe of discourse in order to reach the "truth" of J, that is, the region of the universe where J is true. This interpretation is visibly clear in the Figures above, and appeals to the imagination in a satisfying way, but it has the added benefit of giving fresh meaning to the original name of the shift operator E. Namely, EJ can be read as a proposition that "enlarges" on the meaning of J, in the sense of explaining its practical bearings and clarifying what it means in terms of the available options for differential action and the consequential effects that result from each choice. Treated this way, the enlargement EJ has strong ties to the normal use of J, no matter whether it is understood as a proposition or a function, namely, to act as a figurative device for indicating the models of J, in effect, pointing to the interpretive elements in its fiber of truth J^(-1)(1). It is this kind of "use" that is often compared with the "mention" of a proposition, and thereby hangs a tale. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D53 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Digression: Reflection on Use and Mention | Reflection is turning a topic over in various aspects and in various lights | so that nothing significant about it shall be overlooked -- almost as one | might turn a stone over to see what its hidden side is like or what is | covered by it. | | John Dewey, 'How We Think', [Dew, 57] The contrast drawn in logic between the "use" and the "mention" of a proposition corresponds to the difference that we observe in functional terms between using "J" to indicatet the region J^(-1)(1) and using "J" to indicate the function J. You may think that one of these uses ought to be proscribed, and logicians are quick to prescribe against their confusion. But there seems to be no likelihood in practice that their interactions can be avoided. If the name "J" is used as a sign of the function J, and if the function J has its use in signifying something else, as would constantly be the case when some future theory of signs has given a functional meaning to every sign whatsoever, then is not "J" by transitivity a sign of the thing itself? There are, of course, two answers to this question. Not every act of signifying or referring need be transitive. Not every warrant or guarantee or certificate is automatically transferable, indeed, not many. Not every feature of a feature is a feature of the featuree. Otherwise we have an inference like: If a buffalo is white, and white is a color, then a buffalo is a color. But a buffalo is not, only buff is. The logical or pragmatic distinction between use and mention is cogent and necessary, and so is the analogous functional distinction between determining a value and determining what determines that value, but so are the normal techniques that we use to make these distinctions apply flexibly in practice. The way that the hue and cry about use and mention is raised in logical discussions, you might be led to think that this single dimension of choices embraces the only kinds of use worth mentioning and the only kinds of mention worth using. It will constitute the expeditionary and taxonomic tasks of that future theory of signs to explore and to classify the many other constellations and dimensions of use and mention that are yet to be opened up by the generative potential of full-fledged sign relations. | The well-known capacity that thoughts have -- as doctors have discovered -- | for dissolving and dispersing those hard lumps of deep, ingrowing, morbidly | entangled conflict that arise out of gloomy regions of the self probably rests | on nothing other than their social and worldly nature, which links the individual | being with other people and things; but unfortunately what gives them their power | of healing seems to be the same as what diminishes the quality of personal experience | in them. | | Robert Musil, 'The Man Without Qualities', [Mus, 130] o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D54 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Difference Map of Conjunction | "It doesn't matter what one does", the Man Without Qualities said to himself, | shrugging his shoulders. "In a tangle of forces like this it doesn't make a | scrap of difference." He turned away like a man who has learned renunciation, | almost indeed like a sick man who shrinks from any intensity of contact. And | then, striding through his adjacent dressing-room, he passed a punching-ball | that hung there; he gave it a blow far swifter and harder than is usual in | moods of resignation or states of weakness. | | Robert Musil, 'The Man Without Qualities', [Mus, 8] With the tacit extension map !e!J and the enlargement map EJ well in place, the difference map DJ can be computed along the lines displayed in Table 41, ending up, in this instance, with an expansion of DJ over the cells of [u, v]. Table 41. Computation of DJ (Method 1) o-------------------------------------------------------------------------------o | | | DJ = EJ + !e!J | | | | = J<u + du, v + dv> + J<u, v> | | | | = (u, du)(v, dv) + u v | | | o-------------------------------------------------------------------------------o | | | DJ = 0 | | | | + u v (du) dv + u (v)(du) dv | | | | + u v du (dv) + (u) v du (dv) | | | | + u v du dv + (u)(v) du dv | | | o-------------------------------------------------------------------------------o | | | DJ = u v ((du)(dv)) + u (v)(du) dv + (u) v du (dv) + (u)(v) du dv | | | o-------------------------------------------------------------------------------o Alternatively, the difference map DJ can be expanded over the cells of [du, dv] to arrive at the formulation shown in Table 42. The same development would be obtained from the previous Table by collecting terms in an alternate manner, along the rows rather than the columns of the middle portion of the Table. Table 42. Computation of DJ (Method 2) o-------------------------------------------------------------------------------o | | | DJ = !e!J + EJ | | | | = J<u, v> + J<u + du, v + dv> | | | | = u v + (u, du)(v, dv) | | | | = 0 + u dv + v du + du dv | | | | = 0 + u (du) dv + v du (dv) + ((u, v)) du dv | | | o-------------------------------------------------------------------------------o Even more simply, the same result is reached by matching up the propositional coefficients of !e!J and EJ along the cells of [du, dv] and adding the pairs under boolean sums (that is, "mod 2", where 1 + 1 = 0), as shown in Table 43. Table 43. Computation of DJ (Method 3) o-------------------------------------------------------------------------------o | | | DJ = !e!J + EJ | | | o-------------------------------------------------------------------------------o | | | !e!J = u v (du)(dv) + u v (du) dv + u v du (dv) + u v du dv | | | | EJ = u v (du)(dv) + u (v)(du) dv + (u) v du (dv) + (u)(v) du dv | | | o-------------------------------------------------------------------------------o | | | DJ = 0 . (du)(dv) + u . (du) dv + v . du (dv) + ((u, v)) du dv | | | o-------------------------------------------------------------------------------o o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D55 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Difference Map of Conjunction (cont.) The difference map DJ can also be given a "dispositional" interpretation. First, recall that !e!J exhibits the dispositions to change from anywhere in J to anywhere at all, and EJ enumerates the dispositions to change from anywhere at all to anywhere in J. Next, observe that each of these classes of dispositions may be divided in accordance with the case of J versus (J) that applies to their points of departure and destination, as shown below. Then, since the dispositions corresponding to !e!J and EJ have in common the dispositions to preserve J, their symmetric difference (!e!J, EJ) is made up of all the remaining dispositions, which are in fact disposed to cross the boundary of J in one direction or the other. In other words, we may conclude that DJ expresses the collective disposition to make a definite change with respect to J, no matter what value it holds in the current state of affairs. o-------------------------------------------------------------------------------o | | | !e!J = {Dispositions from J to J } + {Dispositions from J to (J)} | | | | EJ = {Dispositions from J to J } + {Dispositions from (J) to J } | | | | DJ = (!e!J, EJ) | | | | DJ = {Dispositions from J to (J)} + {Dispositions from (J) to J } | | | o-------------------------------------------------------------------------------o o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D56 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Difference Map of Conjunction (concl.) Figures 44-a through 44-d illustrate the difference proposition DJ. o---------------------------------------o | | | o | | / \ | | / \ | | / \ | | o o | | /%\ /%\ | | /%%%\ /%%%\ | | /%%%%%\ /%%%%%\ | | o%%%%%%%o%%%%%%%o | | /%\%%%%%/%\%%%%%/%\ | | /%%%\%%%/%%%\%%%/%%%\ | | /%%%%%\%/%%%%%\%/%%%%%\ | | o%%%%%%%o%%%%%%%o%%%%%%%o | | / \%%%%%/ \%%%%%/ \%%%%%/ \ | | / \%%%/ \%%%/ \%%%/ \ | | / \%/ \%/ \%/ \ | | o o o o o | | |\ / \ /%\ / \ /| | | | \ / \ /%%%\ / \ / | | | | \ / \ /%%%%%\ / \ / | | | | o o%%%%%%%o o | | | | |\ / \%%%%%/ \ /| | | | | | \ / \%%%/ \ / | | | | | u | \ / \%/ \ / | v | | | o---+---o o o---+---o | | | \ / \ / | | | | \ / \ / | | | | du \ / \ / dv | | | o-------o o-------o | | \ / | | \ / | | \ / | | o | | | o---------------------------------------o Figure 44-a. Difference Map of J (Areal) o-----------------------------o | | | o-----o o-----o | | / \ / \ | | / o \ | | / /%\ \ | | o o%%%o o | @ | du |%%%| dv | | /| o o%%%o o | / | \ \%/ / | / | \ o / | / | \ / \ / | / | o-----o o-----o | / | | / o-----------------------------o / o----------------------------------------/----o o-----------------------------o | / | | | | @ | | o-----o o-----o | | | | /%%%%%%%\ / \ | | o---------o o---------o | | /%%%%%%%%%o \ | | / \ / \ | | /%%%%%%%%%/ \ \ | | / o \ | | o%%%%%%%%%o o o | | / /`\ @------\-----------@ |%% du %%%| | dv | | | / /```\ \ | | o%%%%%%%%%o o o | | / /`````\ \ | | \%%%%%%%%%\ / / | | / /```````\ \ | | \%%%%%%%%%o / | | o o`````````o o | | \%%%%%%%/ \ / | | | |````@````| | | | o-----o o-----o | | | |`````\```| | | | | | | |``````\``| | | o-----------------------------o | | u |```````\`| v | | | | |````````\| | | o-----------------------------o | | |`````````| | | | | | | |`````````|\ | | | o-----o o-----o | | o o`````````o \ o | | / \ /%%%%%%%\ | | \ \```````/ \ / | | / o%%%%%%%%%\ | | \ \`````/ \ / | | / / \%%%%%%%%%\ | | \ \```/ \ / | | o o o%%%%%%%%%o | | \ @------\-/---------\---------------@ | du | |%%% dv %%| | | \ o \ / | | o o o%%%%%%%%%o | | \ / \ / | | \ \ /%%%%%%%%%/ | | o---------o o---------o \ | | \ o%%%%%%%%%/ | | \ | | \ / \%%%%%%%/ | | \ | | o-----o o-----o | | \ | | | o----------------------------------------\----o o-----------------------------o \ \ o-----------------------------o \ | | \ | o-----o o-----o | \ | /%%%%%%%\ /%%%%%%%\ | \ | /%%%%%%%%%o%%%%%%%%%\ | \ | /%%%%%%%%%/%\%%%%%%%%%\ | \| o%%%%%%%%%o%%%o%%%%%%%%%o | @ |%% du %%%|%%%|%%% dv %%| | | o%%%%%%%%%o%%%o%%%%%%%%%o | | \%%%%%%%%%\%/%%%%%%%%%/ | | \%%%%%%%%%o%%%%%%%%%/ | | \%%%%%%%/ \%%%%%%%/ | | o-----o o-----o | | | o-----------------------------o Figure 44-b. Difference Map of J (Bundle) o---------------------------------------------------------------------o | | | | | o-------------------o o-------------------o | | / \ / \ | | / o \ | | / / \ \ | | / / \ \ | | / / \ \ | | / / \ \ | | / / \ \ | | o o o o | | | | | | | | | | | | | | | dv .(du) | | du .(dv) | | | | u @<--------------->@<--------------->@ v | | | | | ^ | | | | | | | | | | | | | | | | | | o o | o o | | \ \ | / / | | \ \ | / / | | \ \ | / / | | \ \ | / / | | \ \|/ / | | \ | / | | \ /|\ / | | o-------------------o | o-------------------o | | | | | du . dv | | | | | v | | @ | | | o---------------------------------------------------------------------o Figure 44-c. Difference Map of J (Compact) o-----------------------------------------------------------o | | | u v | | | | @ | | ^^^ | | / | \ | | / | \ | | / | \ | | / | \ | | (du) dv / | \ du (dv) | | / | \ | | / | \ | | / | \ | | / | \ | | v | v | | @ | @ | | u (v) | (u) v | | | | | | | | | | | | | | du | dv | | | | | | | | | | | | | | v | | @ | | | | (u) (v) | | | o-----------------------------------------------------------o Figure 44-d. Difference Map of J (Digraph) o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D57 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Differential of Conjunction | By deploying discourse throughout a calendar, | and by giving a date to each of its elements, | one does not obtain a definitive hierarchy of | precessions and originalities; this hierarchy | is never more than relative to the systems of | discourse that it sets out to evaluate. | | Michel Foucault, 'The Archaeology of Knowledge', [Fou, 143] Finally, at long last, the differential proposition dJ can be gleaned from the difference proposition DJ by ranging over the cells of [u, v] and picking out the linear proposition of [du, dv] that is "closest" to the portion of DJ that touches on each point. The idea of distance that would give this definition unequivocal sense has been referred to in cautionary quotes, the kind we use to distance ourselves from taking a final position. There are obvious notions of approximation that suggest themselves, but finding one that can be justified as ultimately correct is not as straightforward as it seems. | He had drifted into the very heart of the world. | From him to the distant beloved was as far as to | the next tree. | | Robert Musil, 'The Man Without Qualities', [Mus, 144] Let us venture a guess about where these developments might be heading. From the present vantage point, it appears that the ultimate answer to the quandary of distances and the question of a fitting measure may be that, rather than having the constitution of an analytic series depend on our familiar notions of approach, proximity, and approximation, it will be found preferable, and perhaps unavoidable, to turn the tables and let the orders of approximation be defined in terms of our favored and operative notions of formal analysis. Only the aftermath of this conversion, if it does converge, could be hoped to prove whether this hortatory form of analysis and the cohort idea of an analytic form -- the limitary concept of a self-corrective process and the coefficient concept of a completable product -- are truly (in practical reality) the more inceptive and persistent of principles and really (for all practical purposes) the more effective and regulative of ideas. Awaiting that determination, I proceed with what seems like the obvious course, and compute dJ according to the pattern in Table 45. Table 45. Computation of dJ o-------------------------------------------------------------------------------o | | | DJ = u v ((du)(dv)) + u (v)(du) dv + (u) v du (dv) + (u)(v) du dv | | | | => | | | | dj = u v (du, dv) + u (v) dv + (u) v du + (u)(v) . 0 | | | o-------------------------------------------------------------------------------o o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D58 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Differential of Conjunction (concl.) Figures 46-a through 46-d illustrate the proposition dJ, rounded out in our usual array of prospects. This proposition of EU% is what we refer to as the (first order) differential of J, and normally regard as 'the' differential proposition corresponding to J. o---------------------------------------o | | | o | | / \ | | / \ | | / \ | | o o | | /%\ /%\ | | /%%%\ /%%%\ | | /%%%%%\ /%%%%%\ | | o%%%%%%%o%%%%%%%o | | /%\%%%%%/ \%%%%%/%\ | | /%%%\%%%/ \%%%/%%%\ | | /%%%%%\%/ \%/%%%%%\ | | o%%%%%%%o o%%%%%%%o | | / \%%%%%/%\ /%\%%%%%/ \ | | / \%%%/%%%\ /%%%\%%%/ \ | | / \%/%%%%%\ /%%%%%\%/ \ | | o o%%%%%%%o%%%%%%%o o | | |\ / \%%%%%/ \%%%%%/ \ /| | | | \ / \%%%/ \%%%/ \ / | | | | \ / \%/ \%/ \ / | | | | o o o o | | | | |\ / \ / \ /| | | | | | \ / \ / \ / | | | | | u | \ / \ / \ / | v | | | o---+---o o o---+---o | | | \ / \ / | | | | \ / \ / | | | | du \ / \ / dv | | | o-------o o-------o | | \ / | | \ / | | \ / | | o | | | o---------------------------------------o Figure 46-a. Differential of J (Areal) o-----------------------------o | | | o-----o o-----o | | / \ / \ | | / o \ | | / / \ \ | | o o o o | @ | du | | dv | | /| o o o o | / | \ \ / / | / | \ o / | / | \ / \ / | / | o-----o o-----o | / | | / o-----------------------------o / o----------------------------------------/----o o-----------------------------o | / | | | | @ | | o-----o o-----o | | | | /%%%%%%%\ / \ | | o---------o o---------o | | /%%%%%%%%%o \ | | / \ / \ | | /%%%%%%%%%/%\ \ | | / o \ | | o%%%%%%%%%o%%%o o | | / /`\ @------\-----------@ |%% du %%%|%%%| dv | | | / /```\ \ | | o%%%%%%%%%o%%%o o | | / /`````\ \ | | \%%%%%%%%%\%/ / | | / /```````\ \ | | \%%%%%%%%%o / | | o o`````````o o | | \%%%%%%%/ \ / | | | |````@````| | | | o-----o o-----o | | | |`````\```| | | | | | | |``````\``| | | o-----------------------------o | | u |```````\`| v | | | | |````````\| | | o-----------------------------o | | |`````````| | | | | | | |`````````|\ | | | o-----o o-----o | | o o`````````o \ o | | / \ /%%%%%%%\ | | \ \```````/ \ / | | / o%%%%%%%%%\ | | \ \`````/ \ / | | / /%\%%%%%%%%%\ | | \ \```/ \ / | | o o%%%o%%%%%%%%%o | | \ @------\-/---------\---------------@ | du |%%%|%%% dv %%| | | \ o \ / | | o o%%%o%%%%%%%%%o | | \ / \ / | | \ \%/%%%%%%%%%/ | | o---------o o---------o \ | | \ o%%%%%%%%%/ | | \ | | \ / \%%%%%%%/ | | \ | | o-----o o-----o | | \ | | | o----------------------------------------\----o o-----------------------------o \ \ o-----------------------------o \ | | \ | o-----o o-----o | \ | /%%%%%%%\ /%%%%%%%\ | \ | /%%%%%%%%%o%%%%%%%%%\ | \ | /%%%%%%%%%/ \%%%%%%%%%\ | \| o%%%%%%%%%o o%%%%%%%%%o | @ |%% du %%%| |%%% dv %%| | | o%%%%%%%%%o o%%%%%%%%%o | | \%%%%%%%%%\ /%%%%%%%%%/ | | \%%%%%%%%%o%%%%%%%%%/ | | \%%%%%%%/ \%%%%%%%/ | | o-----o o-----o | | | o-----------------------------o Figure 46-b. Differential of J (Bundle) o---------------------------------------------------------------------o | | | | | o-------------------o o-------------------o | | / \ / \ | | / o \ | | / / \ \ | | / / \ \ | | / / \ \ | | / / @ \ \ | | / / ^ ^ \ \ | | o o / \ o o | | | | / \ | | | | | | / \ | | | | | |/ \| | | | | u (du)/ dv du \(dv) v | | | | /| |\ | | | | / | | \ | | | | / | | \ | | | o / o o \ o | | \ / \ / \ / | | \ v \ du dv / v / | | \ @<----------------------->@ / | | \ \ / / | | \ \ / / | | \ o / | | \ / \ / | | o-------------------o o-------------------o | | | | | o---------------------------------------------------------------------o Figure 46-c. Differential of J (Compact) o-----------------------------------------------------------o | | | u v | | @ | | ^ ^ | | / \ | | / \ | | / \ | | / \ | | (du) dv / \ du (dv) | | / \ | | / \ | | / \ | | / \ | | v v | | u (v) @<--------------------->@ (u) v | | du dv | | | | | | | | | | | | | | | | | | | | | | @ | | (u) (v) | | | o-----------------------------------------------------------o Figure 46-d. Differential of J (Digraph) o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D59 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Remainder of Conjunction | I bequeath myself to the dirt to grow from the grass I love, | If you want me again look for me under your bootsoles. | | You will hardly know who I am or what I mean, | But I shall be good health to you nevertheless, | And filter and fibre your blood. | | Failing to fetch me at first keep encouraged, | Missing me one place search another, | I stop some where waiting for you | | Walt Whitman, 'Leaves of Grass', [Whi, 88] Let us now recapitulate the story so far. In effect, we have been carrying out a decomposition of the enlarged proposition EJ in a series of stages. First, we considered the equation EJ = !e!J + DJ, which was involved in the definition of DJ as the difference EJ - !e!J. Next, we contemplated the equation DJ = dJ + rJ, which expresses DJ in terms of two components, the differential dJ that was just extracted and the residual component rJ = DJ - dJ. This remaining proposition rJ can be computed as shown in Table 47. Table 47. Computation of rJ o-------------------------------------------------------------------------------o | | | rJ = DJ + dJ | | | o-------------------------------------------------------------------------------o | | | DJ = u v ((du)(dv)) + u (v)(du) dv + (u) v du (dv) + (u)(v) du dv | | | | dJ = u v (du, dv) + u (v) dv + (u) v du + (u)(v) . 0 | | | o-------------------------------------------------------------------------------o | | | rJ = u v du dv + u (v) du dv + (u) v du dv + (u)(v) du dv | | | o-------------------------------------------------------------------------------o As it happens, the remainder rJ falls under the description of a second order differential rJ = d^2.J. This means that the expansion of EJ in the form: EJ = !e!J + DJ = !e!J + dJ + rJ = d^0.J + d^1.J + d^2.J which is nothing other than the propositional analogue of a Taylor series, is a decomposition that terminates in a finite number of steps. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D60 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Remainder of Conjunction (concl.) Figures 48-a through 48-d illustrate the proposition rJ = d^2.J, which forms the remainder map of J and also, in this instance, the second order differential of J. o---------------------------------------o | | | o | | / \ | | / \ | | / \ | | o o | | / \ / \ | | / \ / \ | | / \ / \ | | o o o | | / \ /%\ / \ | | / \ /%%%\ / \ | | / \ /%%%%%\ / \ | | o o%%%%%%%o o | | / \ /%\%%%%%/%\ / \ | | / \ /%%%\%%%/%%%\ / \ | | / \ /%%%%%\%/%%%%%\ / \ | | o o%%%%%%%o%%%%%%%o o | | |\ / \%%%%%/%\%%%%%/ \ /| | | | \ / \%%%/%%%\%%%/ \ / | | | | \ / \%/%%%%%\%/ \ / | | | | o o%%%%%%%o o | | | | |\ / \%%%%%/ \ /| | | | | | \ / \%%%/ \ / | | | | | u | \ / \%/ \ / | v | | | o---+---o o o---+---o | | | \ / \ / | | | | \ / \ / | | | | du \ / \ / dv | | | o-------o o-------o | | \ / | | \ / | | \ / | | o | | | o---------------------------------------o Figure 48-a. Remainder of J (Areal) o-----------------------------o | | | o-----o o-----o | | / \ / \ | | / o \ | | / /%\ \ | | o o%%%o o | @ | du |%%%| dv | | /| o o%%%o o | / | \ \%/ / | / | \ o / | / | \ / \ / | / | o-----o o-----o | / | | / o-----------------------------o / o----------------------------------------/----o o-----------------------------o | / | | | | @ | | o-----o o-----o | | | | / \ / \ | | o---------o o---------o | | / o \ | | / \ / \ | | / /%\ \ | | / o \ | | o o%%%o o | | / /`\ @------\-----------@ | du |%%%| dv | | | / /```\ \ | | o o%%%o o | | / /`````\ \ | | \ \%/ / | | / /```````\ \ | | \ o / | | o o`````````o o | | \ / \ / | | | |````@````| | | | o-----o o-----o | | | |`````\```| | | | | | | |``````\``| | | o-----------------------------o | | u |```````\`| v | | | | |````````\| | | o-----------------------------o | | |`````````| | | | | | | |`````````|\ | | | o-----o o-----o | | o o`````````o \ o | | / \ / \ | | \ \```````/ \ / | | / o \ | | \ \`````/ \ / | | / /%\ \ | | \ \```/ \ / | | o o%%%o o | | \ @------\-/---------\---------------@ | du |%%%| dv | | | \ o \ / | | o o%%%o o | | \ / \ / | | \ \%/ / | | o---------o o---------o \ | | \ o / | | \ | | \ / \ / | | \ | | o-----o o-----o | | \ | | | o----------------------------------------\----o o-----------------------------o \ \ o-----------------------------o \ | | \ | o-----o o-----o | \ | / \ / \ | \ | / o \ | \ | / /%\ \ | \| o o%%%o o | @ | du |%%%| dv | | | o o%%%o o | | \ \%/ / | | \ o / | | \ / \ / | | o-----o o-----o | | | o-----------------------------o Figure 48-b. Remainder of J (Bundle) o---------------------------------------------------------------------o | | | | | o-------------------o o-------------------o | | / \ / \ | | / o \ | | / / \ \ | | / / \ \ | | / / \ \ | | / / \ \ | | / / \ \ | | o o o o | | | | | | | | | | | | | | | | du dv | | | | | u @<------------------------->@ v | | | | | | | | | | | | | | | | | | | | | o o @ o o | | \ \ ^ / / | | \ \ | / / | | \ \ | / / | | \ \ | / / | | \ \|/ / | | \ du | dv / | | \ /|\ / | | o-------------------o | o-------------------o | | | | | | | | v | | @ | | | o---------------------------------------------------------------------o Figure 48-c. Remainder of J (Compact) o-----------------------------------------------------------o | | | u v | | @ | | ^ | | | | | | | | | | | | | | | | | | | | | | | | | | | | | du | dv | | u (v) @<----------|---------->@ (u) v | | du | dv | | | | | | | | | | | | | | | | | | | | | | | | | | | | | v | | @ | | (u) (v) | | | o-----------------------------------------------------------o Figure 48-d. Remainder of J (Digraph) o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D61 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Summary of Conjunction To establish a convenient reference point for further discussion, Table 49 summarizes the operator actions that have been computed for the form of conjunction, as exemplified by the proposition J. Table 49. Computation Summary for J o-------------------------------------------------------------------------------o | | | !e!J = uv . 1 + u(v) . 0 + (u)v . 0 + (u)(v) . 0 | | | | EJ = uv . (du)(dv) + u(v) . (du)dv + (u)v . du(dv) + (u)(v) . du dv | | | | DJ = uv . ((du)(dv)) + u(v) . (du)dv + (u)v . du(dv) + (u)(v) . du dv | | | | dJ = uv . (du, dv) + u(v) . dv + (u)v . du + (u)(v) . 0 | | | | rJ = uv . du dv + u(v) . du dv + (u)v . du dv + (u)(v) . du dv | | | o-------------------------------------------------------------------------------o o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D62 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Analytic Series: Coordinate Method | And if he is told that something 'is' the way it is, then he thinks: | Well, it could probably just as easily be some other way. So the | sense of possibility might be defined outright as the capacity to | think how everything could "just as easily" be, and to attach no | more importance to what is than to what is not. | | Robert Musil, 'The Man Without Qualities', [Mus, 12] Table 50 exhibits a truth table method for computing the analytic series (or the differential expansion) of a proposition in terms of coordinates. Table 50. Computation of an Analytic Series in Terms of Coordinates o-----------o-------------o-------------oo-------------o---------o-------------o | u v | du dv | u' v' || !e!J EJ | DJ | dJ d^2.J | o-----------o-------------o-------------oo-------------o---------o-------------o | | | || | | | | 0 0 | 0 0 | 0 0 || 0 0 | 0 | 0 0 | | | | || | | | | | 0 1 | 0 1 || 0 | 0 | 0 0 | | | | || | | | | | 1 0 | 1 0 || 0 | 0 | 0 0 | | | | || | | | | | 1 1 | 1 1 || 1 | 1 | 0 1 | | | | || | | | o-----------o-------------o-------------oo-------------o---------o-------------o | | | || | | | | 0 1 | 0 0 | 0 1 || 0 0 | 0 | 0 0 | | | | || | | | | | 0 1 | 0 0 || 0 | 0 | 0 0 | | | | || | | | | | 1 0 | 1 1 || 1 | 1 | 1 0 | | | | || | | | | | 1 1 | 1 0 || 0 | 0 | 1 1 | | | | || | | | o-----------o-------------o-------------oo-------------o---------o-------------o | | | || | | | | 1 0 | 0 0 | 1 0 || 0 0 | 0 | 0 0 | | | | || | | | | | 0 1 | 1 1 || 1 | 1 | 1 0 | | | | || | | | | | 1 0 | 0 0 || 0 | 0 | 0 0 | | | | || | | | | | 1 1 | 0 1 || 0 | 0 | 1 1 | | | | || | | | o-----------o-------------o-------------oo-------------o---------o-------------o | | | || | | | | 1 1 | 0 0 | 1 1 || 1 1 | 0 | 0 0 | | | | || | | | | | 0 1 | 1 0 || 0 | 1 | 1 0 | | | | || | | | | | 1 0 | 0 1 || 0 | 1 | 1 0 | | | | || | | | | | 1 1 | 0 0 || 0 | 1 | 0 1 | | | | || | | | o-----------o-------------o-------------oo-------------o---------o-------------o The first six columns of the Table, taken as a whole, represent the variables of a construct that I describe as the "contingent universe" [u, v, du, dv, u', v'], or the bundle of "contingency spaces" [du, dv, u', v'] over the universe [u, v]. Their placement to the left of the double bar indicates that all of them amount to independent variables, but there is a co-dependency among them, as described by the following equations: o-------------------------------------------------o | | | u' = u + du = (u, du) | | | | v' = v + du = (v, dv) | | | o-------------------------------------------------o These relations correspond to the formal substitutions that are made in defining EJ and DJ. For now, the whole rigamarole of contingency spaces can be regarded as a technical device for achieving the effect of these substitutions, adapted to a setting where functional compositions and other symbolic manipulations are difficult to contemplate and execute. The five columns to the right of the double bar in Table 50 contain the values of the dependent variables {!e!J, EJ, DJ, dJ, d^2.J}. These are normally interpreted as values of functions WJ : EU -> B or as values of propositions in the extended universe [u, v, du, dv], but the dependencies prevailing in the contingent universe make it possible to regard these same final values as arising via functions on alternative lists of arguments, say, <u, v, u', v'>. The column for !e!J is computed as J<u, v> = uv. This, along with the columns for u and v, illustrates the Table's "structure-sharing" scheme, listing only the initial entries of each constant block. The column for EJ is computed by means of the following chain of identities, where the contingent variables u' and v' are defined as u' = u + du and v' = v + dv. o--------------------------------------------------------------o | | | EJ<u, v, du, dv> = J<u + du, v + dv> = J<u', v'> | | | o--------------------------------------------------------------o This makes it easy to determine EJ by inspection, computing the conjunction J<u', v'> = u' v' from the columns headed u' and v'. Since all of these forms express the same proposition EJ in EU%, the dependence on du and dv is still present but merely left implicit in the final variant J<u', v'>. NB. On occasion, it is tempting to use the further notation J'<u, v> = J<u', v'>, especially to suggest a transformation that acts on whole propositions, for example, taking the proposition J into the proposition J' = EJ. The prime ['] then signifies an action that is mediated by a field of choices, namely, the values that are picked out for the contingent variables in sweeping through the initial universe. But this heaps an unwieldy lot of construed intentions on a rather slight character, and puts too high a premium on the constant correctness of its interpretation. In practice, therefore, it is best to avoid this usage. Given the values of !e!J and EJ, the columns for the remaining functions can be filled in quickly. The difference map is computed according to the relation DJ = !e!J + EJ. The first order differential dJ is found by looking in each block of constant <u, v> and choosing the linear function of <du, dv> that best approximates DJ in that block. Finally, the remainder is computed as rJ = DJ + dJ, in this case yielding the second order differential d^2.J. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D63 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Analytic Series: Recap Let us now summarize the results of Table 50 by writing down for each column, and for each block of constant <u, v>, a reasonably canonical symbolic expression for the function of <du, dv> that appears there. The synopsis formed in this way is presented in Table 51. As one has a right to expect, it confirms the results that were obtained previously by operating solely in terms of the formal calculus. Table 51. Computation of an Analytic Series in Symbolic Terms o-----------o---------o------------o------------o------------o-----------o | u v | J | EJ | DJ | dJ | d^2.J | o-----------o---------o------------o------------o------------o-----------o | | | | | | | | 0 0 | 0 | du dv | du dv | () | du dv | | | | | | | | | 0 1 | 0 | du (dv) | du (dv) | du | du dv | | | | | | | | | 1 0 | 0 | (du) dv | (du) dv | dv | du dv | | | | | | | | | 1 1 | 1 | (du)(dv) | ((du)(dv)) | (du, dv) | du dv | | | | | | | | o-----------o---------o------------o------------o------------o-----------o Figures 52 and 53 provide a quick overview of the analysis performed so far, giving the successive decompositions of EJ = J + DJ and DJ = dJ + rJ in two different styles of diagram. o o o /%\ /%\ / \ /%%%\ /%%%\ / \ o%%%%%o o%%%%%o o o / \%%%/ \ /%\%%%/%\ /%\ /%\ / \%/ \ /%%%\%/%%%\ /%%%\ /%%%\ o o o o%%%%%o%%%%%o o%%%%%o%%%%%o /%\ / \ /%\ / \%%%/%\%%%/ \ /%\%%%/%\%%%/%\ /%%%\ / \ /%%%\ / \%/%%%\%/ \ /%%%\%/%%%\%/%%%\ o%%%%%o o%%%%%o o o%%%%%o o o%%%%%o%%%%%o%%%%%o / \%%%/ \ / \%%%/ \ / \ / \%%%/ \ / \ / \%%%/ \%%%/ \%%%/ \ / \%/ \ / \%/ \ / \ / \%/ \ / \ / \%/ \%/ \%/ \ o o o o o o o o o o o o o o o |\ / \ /%\ / \ /| |\ / \ / \ / \ /| |\ / \ /%\ / \ /| | \ / \ /%%%\ / \ / | | \ / \ / \ / \ / | | \ / \ /%%%\ / \ / | | o o%%%%%o o | | o o o o | | o o%%%%%o o | | |\ / \%%%/ \ /| | | |\ / \ / \ /| | | |\ / \%%%/ \ /| | |u | \ / \%/ \ / | v| |u | \ / \ / \ / | v| |u | \ / \%/ \ / | v| o--+--o o o--+--o o--+--o o o--+--o o--+--o o o--+--o | \ / \ / | | \ / \ / | | \ / \ / | | du \ / \ / dv | | du \ / \ / dv | | du \ / \ / dv | o-----o o-----o o-----o o-----o o-----o o-----o \ / \ / \ / \ / \ / \ / o o o EJ = J + DJ o-----------------------o o-----------------------o o-----------------------o | | | | | | | o--o o--o | | o--o o--o | | o--o o--o | | / \ / \ | | / \ / \ | | / \ / \ | | / o \ | | / o \ | | / o \ | | / u / \ v \ | | / u / \ v \ | | / u / \ v \ | | o /->-\ o | | o /->-\ o | | o / \ o | | | o \ / o | | | | o \ / o | | | | o o | | | | @--|->@<-|--@ | | | | @<-|--@--|->@ | | | | @<-|->@<-|->@ | | | | o ^ o | | | | o | o | | | | o ^ o | | | o \ | / o | | o \ | / o | | o \ | / o | | \ \|/ / | | \ \|/ / | | \ \|/ / | | \ | / | | \ | / | | \ | / | | \ /|\ / | | \ /|\ / | | \ /|\ / | | o--o | o--o | | o--o v o--o | | o--o v o--o | | @ | | @ | | @ | o-----------------------o o-----------------------o o-----------------------o Figure 52. Decomposition of the Enlarged Conjunction EJ = (J, DJ) o o o / \ / \ / \ / \ / \ / \ o o o o o o /%\ /%\ /%\ /%\ / \ / \ /%%%\ /%%%\ /%%%\%/%%%\ / \ / \ o%%%%%o%%%%%o o%%%%%o%%%%%o o o o /%\%%%/%\%%%/%\ /%\%%%/ \%%%/%\ / \ /%\ / \ /%%%\%/%%%\%/%%%\ /%%%\%/ \%/%%%\ / \ /%%%\ / \ o%%%%%o%%%%%o%%%%%o o%%%%%o o%%%%%o o o%%%%%o o / \%%%/ \%%%/ \%%%/ \ / \%%%/%\ /%\%%%/ \ / \ /%\%%%/%\ / \ / \%/ \%/ \%/ \ / \%/%%%\ /%%%\%/ \ / \ /%%%\%/%%%\ / \ o o o o o o o%%%%%o%%%%%o o o o%%%%%o%%%%%o o |\ / \ /%\ / \ /| |\ / \%%%/ \%%%/ \ /| |\ / \%%%/%\%%%/ \ /| | \ / \ /%%%\ / \ / | | \ / \%/ \%/ \ / | | \ / \%/%%%\%/ \ / | | o o%%%%%o o | | o o o o | | o o%%%%%o o | | |\ / \%%%/ \ /| | | |\ / \ / \ /| | | |\ / \%%%/ \ /| | |u | \ / \%/ \ / | v| |u | \ / \ / \ / | v| |u | \ / \%/ \ / | v| o--+--o o o--+--o o--+--o o o--+--o o--+--o o o--+--o | \ / \ / | | \ / \ / | | \ / \ / | | du \ / \ / dv | | du \ / \ / dv | | du \ / \ / dv | o-----o o-----o o-----o o-----o o-----o o-----o \ / \ / \ / \ / \ / \ / o o o DJ = dJ + ddJ o-----------------------o o-----------------------o o-----------------------o | | | | | | | o--o o--o | | o--o o--o | | o--o o--o | | / \ / \ | | / \ / \ | | / \ / \ | | / o \ | | / o \ | | / o \ | | / u / \ v \ | | / u / \ v \ | | / u / \ v \ | | o / \ o | | o / \ o | | o / \ o | | | o o | | | | o o | | | | o o | | | | @<-|->@<-|->@ | | | | @<-|->@<-|->@ | | | | @<-|-----|->@ | | | | o ^ o | | | | ^ o o ^ | | | | o @ o | | | o \ | / o | | o \ \ / / o | | o \ ^ / o | | \ \|/ / | | \ --\-/-- / | | \ \|/ / | | \ | / | | \ o / | | \ | / | | \ /|\ / | | \ / \ / | | \ /|\ / | | o--o v o--o | | o--o o--o | | o--o v o--o | | @ | | @ | | @ | o-----------------------o o-----------------------o o-----------------------o Figure 53. Decomposition of the Differed Conjunction DJ = (dJ, ddJ) o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D64 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Terminological Interlude | Lastly, my attention was especially attracted, not so much to the scene, | as to the mirrors that produced it. These mirrors were broken in parts. | Yes, they were marked and scratched; they had been "starred", in spite | of their solidity ... | | Gaston Leroux, 'The Phantom of the Opera', [Ler, 230] At this point several issues of terminology have accrued enough substance to intrude on our discussion. The remarks of this Section are intended to accomplish two goals. First, I call attention to important aspects of the previous series of Figures, translating into literal terms what they depict in iconic forms, and I restress the most important structural elements that they indicate. Next, I prepare the way for taking on more complex examples of transformations, whose target universes have more than a single dimension. In talking about the actions of operators it is important to keep in mind the distinctions between the operators per se, their operands, and their results. Furthermore, in working with composite forms of operators W = <W_1, ..., W_n>, transformations F = <F_1, ..., F_n>, and target domains X% = [x_1, ..., x_n], we need to preserve a clear distinction between the compound entity of each given type and any one of its separate components. It is curious, given the usefulness of the concepts "operator" and "operand", that we seem to lack a generic term, formed on the same root, for the corresponding result of an operation. Following the obvious paradigm would lead on to words like "opus", "opera", and "operant", but these words are too affected with clang associations to work well at present, though they might be adapted in time. One current usage gets around this problem by using the substantive "map" as a systematic epithet to express the result of each operator's action. I am following this practice as far as possible, for example, using the phrase "tangent map" to denote the end product of the tangent functor acting on its operand map. Scholium. See [JGH, 6-9] for a good account of tangent functors and tangent maps in ordinary analysis, and for examples of their use in mechanics. This work as a whole is a model of clarity in applying functorial principles to problems in physical dynamics. Whenever we focus on isolated propositions, on single components of composite operators, or on the portions of transformations that have 1-dimensional ranges, we are free to shift between the native form of a proposition J : U -> B and the thematized form of a mapping J : U% -> [x] without much trouble. In these cases we are able to tolerate a higher degree of ambiguity about the precise nature of the input and output domains of an operator than we otherwise might. For example, in the preceding treatment of the example J, and for each operator W in the set {!e!, !h!, E, D, d, r}, both the operand J and the result WJ could be viewed in either one of two ways. On the one hand, we could regard them as propositions J : U -> B and WJ : EU -> B, ignoring the qualitative distinction between the range [x] ~=~ B of !e!J and the range [dx] ~=~ D of the other WJ's. This is what we usually do when we content ourselves simply with coloring in regions of venn diagrams. On the other hand, we could view these entities as maps J : U% -> [x] = X% and !e!J : EU% -> [x] c EX% or WJ : EU% -> [dx] c EX%, in which case the qualitative characters of the output features are not allowed to go without saying, nor thus at the risk of being forgotten. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D65 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Terminological Interlude (cont.) At the beginning of this Division I recast the natural form of a proposition J : U -> B into the thematic role of a transformation J : U% -> [x], where x was a variable recruited to express the newly independent ¢(J). However, in my computations and representations of operator actions I immediately lapsed back to viewing the results as native elements of the extended universe EU%, in other words, as propositions WJ : EU -> B, where W ranged over the set {!e!, E, D, d, r}. That is as it should be. In fact, I have worked hard to devise a language that gives us all of these competing advantages, the flexibility to exchange terms and types that bear equal information value, and the capacity to reflect as quickly and as wittingly as a controlled reflex on the fibers of our propositions, independently of whether they express amusements, beliefs, or conjectures. As we take on target spaces of increasing dimension, however, these types of confusions (and confusions of types) become less and less permissible. For this reason, Tables 54 and 55 present a rather detailed summary of the notation and the terminology that I am using here, as applied to the case of J = uv. The rationale of these Tables is not so much to train more elephant guns on this poor drosophila of an example, but to establish the general paradigm with enough solidity to bear the weight of abstraction that is coming on down the road. Table 54 provides basic notation and descriptive information for the objects and operators that are used used in this Example, giving the generic type (or broadest defined type) for each entity. Here, the sans serif operators, $W$ in {$e$, $E$, $D$, $d$, $r$}, and their components W in {!e!, !h!, E, D, d, r} both have the same broad type $W$, W : (U% -> X%) -> (EU% -> EX%), as would be expected of operators that map transformations J : U% -> X% to extended transformations $W$J, WJ : EU% -> EX%. Table 54. Cast of Characters: Expansive Subtypes of Objects and Operators o------o-------------------------o------------------o----------------------------o | Item | Notation | Description | Type | o------o-------------------------o------------------o----------------------------o | | | | | | U% | = [u, v] | Source Universe | [B^2] | | | | | | o------o-------------------------o------------------o----------------------------o | | | | | | X% | = [x] | Target Universe | [B^1] | | | | | | o------o-------------------------o------------------o----------------------------o | | | | | | EU% | = [u, v, du, dv] | Extended | [B^2 x D^2] | | | | Source Universe | | | | | | | o------o-------------------------o------------------o----------------------------o | | | | | | EX% | = [x, dx] | Extended | [B^1 x D^1] | | | | Target Universe | | | | | | | o------o-------------------------o------------------o----------------------------o | | | | | | J | J : U -> B | Proposition | (B^2 -> B) c [B^2] | | | | | | o------o-------------------------o------------------o----------------------------o | | | | | | J | J : U% -> X% | Transformation, | [B^2] -> [B^1] | | | | or Mapping | | | | | | | o------o-------------------------o------------------o----------------------------o | | | | | | W | W : | Operator | | | | U% -> EU%, | | [B^2] -> [B^2 x D^2], | | | X% -> EX%, | | [B^1] -> [B^1 x D^1], | | | (U%->X%)->(EU%->EX%), | | ([B^2] -> [B^1]) | | | for each W among: | | -> | | | e!, !h!, E, D, d | | ([B^2 x D^2]->[B^1 x D^1]) | | | | | | o------o-------------------------o------------------o----------------------------o | | | | | !e! | | Tacit Extension Operator !e! | | !h! | | Trope Extension Operator !h! | | E | | Enlargement Operator E | | D | | Difference Operator D | | d | | Differential Operator d | | | | | o------o-------------------------o------------------o----------------------------o | | | | | | $W$ | $W$ : | Operator | | | | U% -> $T$U% = EU%, | | [B^2] -> [B^2 x D^2], | | | X% -> $T$X% = EX%, | | [B^1] -> [B^1 x D^1], | | | (U%->X%)->($T$U%->$T$X%)| | ([B^2] -> [B^1]) | | | for each $W$ among: | | -> | | | $e$, $E$, $D$, $T$ | | ([B^2 x D^2]->[B^1 x D^1]) | | | | | | o------o-------------------------o------------------o----------------------------o | | | | | $e$ | | Radius Operator $e$ = <!e!, !h!> | | $E$ | | Secant Operator $E$ = <!e!, E > | | $D$ | | Chord Operator $D$ = <!e!, D > | | $T$ | | Tangent Functor $T$ = <!e!, d > | | | | | o------o-------------------------o-----------------------------------------------o o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D66 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Terminological Interlude (concl.) Table 55 supplies a more detailed outline of terminology for operators and their results. Here, I list the restrictive subtype (or narrowest defined subtype) that applies to each entity, and I indicate across the span of the Table the whole spectrum of alternative types that color the interpretation of each symbol. Accordingly, each of the component operator maps WJ, since their ranges are 1-dimensional (of type B^1 or D^1), can be regarded either as propositions WJ : EU -> B or as logical transformations WJ : EU% -> X%. As a rule, the plan of the Table allows us to name each entry by detaching the adjective at the left of its row and prefixing it to the generic noun at the top of its column. In one case, however, it is customary to depart from this scheme. Because the phrase "differential proposition", applied to the result dJ : EU -> D, does not distinguish it from the general run of differential propositions G : EU -> B, it is usual to single out dJ as the "tangent proposition" of J. Table 55. Synopsis of Terminology: Restrictive and Alternative Subtypes o--------------o----------------------o--------------------o----------------------o | | Operator | Proposition | Map | o--------------o----------------------o--------------------o----------------------o | | | | | | Tacit | !e! : | !e!J : | !e!J : | | Extension | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> B | [u,v,du,dv]->[x] | | | (U%->X%)->(EU%->X%) | B^2 x D^2 -> B | [B^2 x D^2]->[B^1] | | | | | | o--------------o----------------------o--------------------o----------------------o | | | | | | Trope | !h! : | !h!J : | !h!J : | | Extension | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx] | | | (U%->X%)->(EU%->dX%) | B^2 x D^2 -> D | [B^2 x D^2]->[D^1] | | | | | | o--------------o----------------------o--------------------o----------------------o | | | | | | Enlargement | E : | EJ : | EJ : | | Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx] | | | (U%->X%)->(EU%->dX%) | B^2 x D^2 -> D | [B^2 x D^2]->[D^1] | | | | | | o--------------o----------------------o--------------------o----------------------o | | | | | | Difference | D : | DJ : | DJ : | | Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx] | | | (U%->X%)->(EU%->dX%) | B^2 x D^2 -> D | [B^2 x D^2]->[D^1] | | | | | | o--------------o----------------------o--------------------o----------------------o | | | | | | Differential | d : | dJ : | dJ : | | Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx] | | | (U%->X%)->(EU%->dX%) | B^2 x D^2 -> D | [B^2 x D^2]->[D^1] | | | | | | o--------------o----------------------o--------------------o----------------------o | | | | | | Remainder | r : | rJ : | rJ : | | Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx] | | | (U%->X%)->(EU%->dX%) | B^2 x D^2 -> D | [B^2 x D^2]->[D^1] | | | | | | o--------------o----------------------o--------------------o----------------------o | | | | | | Radius | $e$ = <!e!, !h!> : | | $e$J : | | Operator | U%->EU%, X%->EX%, | | [u,v,du,dv]->[x, dx] | | | (U%->X%)->(EU%->EX%) | | [B^2 x D^2]->[B x D] | | | | | | o--------------o----------------------o--------------------o----------------------o | | | | | | Secant | $E$ = <!e!, E> : | | $E$J : | | Operator | U%->EU%, X%->EX%, | | [u,v,du,dv]->[x, dx] | | | (U%->X%)->(EU%->EX%) | | [B^2 x D^2]->[B x D] | | | | | | o--------------o----------------------o--------------------o----------------------o | | | | | | Chord | $D$ = <!e!, D> : | | $D$J : | | Operator | U%->EU%, X%->EX%, | | [u,v,du,dv]->[x, dx] | | | (U%->X%)->(EU%->EX%) | | [B^2 x D^2]->[B x D] | | | | | | o--------------o----------------------o--------------------o----------------------o | | | | | | Tangent | $T$ = <!e!, d> : | dJ : | $T$J : | | Functor | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[x, dx] | | | (U%->X%)->(EU%->EX%) | B^2 x D^2 -> D | [B^2 x D^2]->[B x D] | | | | | | o--------------o----------------------o--------------------o----------------------o o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D67 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o End of Perfunctory Chatter: Time to Roll the Clip! Two steps remain to finish the analysis of J that I began so long ago. First, I need to paste the accumulated heap of flat pictures into the frames of transformations, filling out the shapes of the operator maps $W$J : EU% -> EX%. This scheme is executed in two styles, using the "areal views" in Figures 56-a and the "box views" in Figures 56-b. Finally, in Figures 57-1 to 57-4 I put all the pieces together to construct the full operator diagrams for $W$ : J -> $W$J. There is a large amount of redundancy in these three series of figures. At this early stage of exposition I thought that it would be better not to tax the reader's imagination, and to guarantee that the author, at least, has worked through the relevant exercises. I hope the reader will excuse the flagrant use of space and try to view these snapshots as successive frames in the animation of logic that they are meant to become. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D68 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Operator Maps: Areal Views o /X\ /XXX\ oXXXXXo /X\XXX/X\ /XXX\X/XXX\ oXXXXXoXXXXXo / \XXX/X\XXX/ \ / \X/XXX\X/ \ o oXXXXXo o / \ / \XXX/ \ / \ / \ / \X/ \ / \ o o o o o =|\ / \ / \ / \ /|= = | \ / \ / \ / \ / | = = | o o o o | = = | |\ / \ / \ /| | = = |u | \ / \ / \ / | v| = o o--+--o o o--+--o o //\ | \ / \ / | /\\ ////\ | du \ / \ / dv | /\\\\ o/////o o-----o o-----o o\\\\\o //\/////\ \ / /\\\\\/\\ ////\/////\ \ / /\\\\\/\\\\ o/////o/////o o o\\\\\o\\\\\o / \/////\//// \ = = / \\\\/\\\\\/ \ / \/////\// \ = = / \\/\\\\\/ \ o o/////o o = = o o\\\\\o o / \ / \//// \ / \ = = / \ / \\\\/ \ / \ / \ / \// \ / \ = = / \ / \\/ \ / \ o o o o o o o o o o |\ / \ / \ / \ /| |\ / \ / \ / \ /| | \ / \ / \ / \ / | | \ / \ / \ / \ / | | o o o o | | o o o o | | |\ / \ / \ /| | | |\ / \ / \ /| | |u | \ / \ / \ / | v| |u | \ / \ / \ / | v| o--+--o o o--+--o o o--+--o o o--+--o . | \ / \ / | /X\ | \ / \ / | . .| du \ / \ / dv | /XXX\ | du \ / \ / dv |. o-----o o-----o /XXXXX\ o-----o o-----o . \ / /XXXXXXX\ \ / . . \ / /XXXXXXXXX\ \ / . . o oXXXXXXXXXXXo o . . //\XXXXXXXXX/\\ . . ////\XXXXXXX/\\\\ . !e!J //////\XXXXX/\\\\\\ !h!J . ////////\XXX/\\\\\\\\ . . //////////\X/\\\\\\\\\\ . . o///////////o\\\\\\\\\\\o . . |\////////// \\\\\\\\\\/| . . | \//////// \\\\\\\\/ | . . | \////// \\\\\\/ | . . | \//// \\\\/ | . .| x \// \\/ dx |. o-----o o-----o \ / \ / x = uv \ / dx = uv \ / \ / o Figure 56-a1. Radius Map of the Conjunction J = uv o /X\ /XXX\ oXXXXXo //\XXX//\ ////\X////\ o/////o/////o /\\/////\////\\ /\\\\/////\//\\\\ o\\\\\o/////o\\\\\o / \\\\/ \//// \\\\/ \ / \\/ \// \\/ \ o o o o o =|\ / \ /\\ / \ /|= = | \ / \ /\\\\ / \ / | = = | o o\\\\\o o | = = | |\ / \\\\/ \ /| | = = |u | \ / \\/ \ / | v| = o o--+--o o o--+--o o //\ | \ / \ / | /\\ ////\ | du \ / \ / dv | /\\\\ o/////o o-----o o-----o o\\\\\o //\/////\ \ / / \\\\/ \ ////\/////\ \ / / \\/ \ o/////o/////o o o o o / \/////\//// \ = = /\\ / \ /\\ / \/////\// \ = = /\\\\ / \ /\\\\ o o/////o o = = o\\\\\o o\\\\\o / \ / \//// \ / \ = = / \\\\/ \ / \\\\/ \ / \ / \// \ / \ = = / \\/ \ / \\/ \ o o o o o o o o o o |\ / \ / \ / \ /| |\ / \ /\\ / \ /| | \ / \ / \ / \ / | | \ / \ /\\\\ / \ / | | o o o o | | o o\\\\\o o | | |\ / \ / \ /| | | |\ / \\\\/ \ /| | |u | \ / \ / \ / | v| |u | \ / \\/ \ / | v| o--+--o o o--+--o o o--+--o o o--+--o . | \ / \ / | /X\ | \ / \ / | . .| du \ / \ / dv | /XXX\ | du \ / \ / dv |. o-----o o-----o /XXXXX\ o-----o o-----o . \ / /XXXXXXX\ \ / . . \ / /XXXXXXXXX\ \ / . . o oXXXXXXXXXXXo o . . //\XXXXXXXXX/\\ . . ////\XXXXXXX/\\\\ . !e!J //////\XXXXX/\\\\\\ EJ . ////////\XXX/\\\\\\\\ . . //////////\X/\\\\\\\\\\ . . o///////////o\\\\\\\\\\\o . . |\////////// \\\\\\\\\\/| . . | \//////// \\\\\\\\/ | . . | \////// \\\\\\/ | . . | \//// \\\\/ | . .| x \// \\/ dx |. o-----o o-----o \ / \ / dx = (u, du)(v, dv) x = uv \ / \ / dx = uv + u dv + v du + du dv \ / o Figure 56-a2. Secant Map of the Conjunction J = uv o //\ ////\ o/////o /X\////X\ /XXX\//XXX\ oXXXXXoXXXXXo /\\XXX/X\XXX/\\ /\\\\X/XXX\X/\\\\ o\\\\\oXXXXXo\\\\\o / \\\\/ \XXX/ \\\\/ \ / \\/ \X/ \\/ \ o o o o o =|\ / \ /\\ / \ /|= = | \ / \ /\\\\ / \ / | = = | o o\\\\\o o | = = | |\ / \\\\/ \ /| | = = |u | \ / \\/ \ / | v| = o o--+--o o o--+--o o //\ | \ / \ / | / \ ////\ | du \ / \ / dv | / \ o/////o o-----o o-----o o o //\/////\ \ / /\\ /\\ ////\/////\ \ / /\\\\ /\\\\ o/////o/////o o o\\\\\o\\\\\o / \/////\//// \ = = /\\\\\/\\\\\/\\ / \/////\// \ = = /\\\\\/\\\\\/\\\\ o o/////o o = = o\\\\\o\\\\\o\\\\\o / \ / \//// \ / \ = = / \\\\/ \\\\/ \\\\/ \ / \ / \// \ / \ = = / \\/ \\/ \\/ \ o o o o o o o o o o |\ / \ / \ / \ /| |\ / \ /\\ / \ /| | \ / \ / \ / \ / | | \ / \ /\\\\ / \ / | | o o o o | | o o\\\\\o o | | |\ / \ / \ /| | | |\ / \\\\/ \ /| | |u | \ / \ / \ / | v| |u | \ / \\/ \ / | v| o--+--o o o--+--o o o--+--o o o--+--o . | \ / \ / | /X\ | \ / \ / | . .| du \ / \ / dv | /XXX\ | du \ / \ / dv |. o-----o o-----o /XXXXX\ o-----o o-----o . \ / /XXXXXXX\ \ / . . \ / /XXXXXXXXX\ \ / . . o oXXXXXXXXXXXo o . . //\XXXXXXXXX/\\ . . ////\XXXXXXX/\\\\ . !e!J //////\XXXXX/\\\\\\ DJ . ////////\XXX/\\\\\\\\ . . //////////\X/\\\\\\\\\\ . . o///////////o\\\\\\\\\\\o . . |\////////// \\\\\\\\\\/| . . | \//////// \\\\\\\\/ | . . | \////// \\\\\\/ | . . | \//// \\\\/ | . .| x \// \\/ dx |. o-----o o-----o \ / \ / dx = (u, du)(v, dv) - uv x = uv \ / \ / dx = u dv + v du + du dv \ / o Figure 56-a3. Chord Map of the Conjunction J = uv o //\ ////\ o/////o /X\////X\ /XXX\//XXX\ oXXXXXoXXXXXo /\\XXX//\XXX/\\ /\\\\X////\X/\\\\ o\\\\\o/////o\\\\\o / \\\\/\\////\\\\\/ \ / \\/\\\\//\\\\\/ \ o o\\\\\o\\\\\o o =|\ / \\\\/ \\\\/ \ /|= = | \ / \\/ \\/ \ / | = = | o o o o | = = | |\ / \ / \ /| | = = |u | \ / \ / \ / | v| = o o--+--o o o--+--o o //\ | \ / \ / | / \ ////\ | du \ / \ / dv | / \ o/////o o-----o o-----o o o //\/////\ \ / /\\ /\\ ////\/////\ \ / /\\\\ /\\\\ o/////o/////o o o\\\\\o\\\\\o / \/////\//// \ = = /\\\\\/ \\\\/\\ / \/////\// \ = = /\\\\\/ \\/\\\\ o o/////o o = = o\\\\\o o\\\\\o / \ / \//// \ / \ = = / \\\\/\\ /\\\\\/ \ / \ / \// \ / \ = = / \\/\\\\ /\\\\\/ \ o o o o o o o\\\\\o\\\\\o o |\ / \ / \ / \ /| |\ / \\\\/ \\\\/ \ /| | \ / \ / \ / \ / | | \ / \\/ \\/ \ / | | o o o o | | o o o o | | |\ / \ / \ /| | | |\ / \ / \ /| | |u | \ / \ / \ / | v| |u | \ / \ / \ / | v| o--+--o o o--+--o o o--+--o o o--+--o . | \ / \ / | /X\ | \ / \ / | . .| du \ / \ / dv | /XXX\ | du \ / \ / dv |. o-----o o-----o /XXXXX\ o-----o o-----o . \ / /XXXXXXX\ \ / . . \ / /XXXXXXXXX\ \ / . . o oXXXXXXXXXXXo o . . //\XXXXXXXXX/\\ . . ////\XXXXXXX/\\\\ . !e!J //////\XXXXX/\\\\\\ dJ . ////////\XXX/\\\\\\\\ . . //////////\X/\\\\\\\\\\ . . o///////////o\\\\\\\\\\\o . . |\////////// \\\\\\\\\\/| . . | \//////// \\\\\\\\/ | . . | \////// \\\\\\/ | . . | \//// \\\\/ | . .| x \// \\/ dx |. o-----o o-----o \ / \ / x = uv \ / dx = u dv + v du \ / \ / o Figure 56-a4. Tangent Map of the Conjunction J = uv o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D69 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Operator Maps: Box Views o-----------------------o | | | | | | | o--o o--o | | / \ / \ | | / o \ | | / du / \ dv \ | | o / \ o | | | o o | | | | | | | | | | o o | | | o \ / o | | \ \ / / | | \ o / | | \ / \ / | | o--o o--o | | | | | | | o-----------------------@ \ o-----------------------o \ | | \ | | \ | | \ | o--o o--o | \ | / \ / \ | \ | / o \ | \ | / du / \ dv \ | \ | o / \ o | \ | | o o | @ \ | | | | | |\ \ | | o o | | \ \ | o \ / o | \ \ | \ \ / / | \ \ | \ o / | \ \ | \ / \ / | \ \ | o--o o--o | \ \ | | \ \ | | \ \ | | \ \ o-----------------------o \ \ \ \ o-----------------------@ o--------\----------\---o o-----------------------o | |\ | \ \ | |```````````````````````| | | \ | \ @ | |```````````````````````| | | \| \ | |```````````````````````| | o--o o--o | \ o--o \o--o | |``````o--o```o--o``````| | / \ / \ | |\ / \ /\ \ | |`````/````\`/````\`````| | / o \ | | \ / o @ \ | |````/``````o``````\````| | / du / \ dv \ | | \/ du /`\ dv \ | |```/``du``/`\``dv``\```| | o / \ o | | o\ /```\ o | |``o``````/```\``````o``| | | o o | | | | \ o`````o | | |``|`````o`````o`````|``| | | | | | | | | @ |``@--|-----|------@``|`````|`````|`````|``| | | o o | | | | o`````o | | |``|`````o`````o`````|``| | o \ / o | | o \```/ o | |``o``````\```/``````o``| | \ \ / / | | \ \`/ / | |```\``````\`/``````/```| | \ o / | | \ o / | |````\``````o``````/````| | \ / \ / | | \ / \ / | |`````\````/`\````/`````| | o--o o--o | | o--o o--o | |``````o--o```o--o``````| | | | | |```````````````````````| | | | | |```````````````````````| | | | | |```````````````````````| o-----------------------o o-----------------------o o-----------------------o \ / \ / \ / \ !h!J / \ J / \ !h!J / \ / \ / \ / \ / o----------\---------/----------o \ / \ / | \ / | \ / \ / | \ / | \ / \ / | o-----o-----o | \ / \ / | /`````````````\ | \ / \ / | /```````````````\ | \ / o------\---/------o | /`````````````````\ | o------\---/------o | \ / | | /```````````````````\ | | \ / | | o--o--o | | /`````````````````````\ | | o--o--o | | /```````\ | | o```````````````````````o | | /```````\ | | /`````````\ | | |```````````````````````| | | /`````````\ | | o```````````o | | |```````````````````````| | | o```````````o | | |````dx`````| @----@ |```````````x`````@-----|------@ |``` dx ````| | | o```````````o | | |```````````````````````| | | o```````````o | | \`````````/ | | |```````````````````````| | | \`````````/ | | \```````/ | | o```````````````````````o | | \```````/ | | o-----o | | \`````````````````````/ | | o-----o | | | | \```````````````````/ | | | o-----------------o | \`````````````````/ | o-----------------o | \```````````````/ | | \`````````````/ | | o-----------o | | | | | o-------------------------------o Figure 56-b1. Radius Map of the Conjunction J = uv o-----------------------o | | | | | | | o--o o--o | | / \ / \ | | / o \ | | / du /`\ dv \ | | o /```\ o | | | o`````o | | | | |`````| | | | | o`````o | | | o \```/ o | | \ \`/ / | | \ o / | | \ / \ / | | o--o o--o | | | | | | | o-----------------------@ \ o-----------------------o \ | | \ | | \ | | \ | o--o o--o | \ | /````\ / \ | \ | /``````o \ | \ | /``du``/ \ dv \ | \ | o``````/ \ o | \ | |`````o o | @ \ | |`````| | | |\ \ | |`````o o | | \ \ | o``````\ / o | \ \ | \``````\ / / | \ \ | \``````o / | \ \ | \````/ \ / | \ \ | o--o o--o | \ \ | | \ \ | | \ \ | | \ \ o-----------------------o \ \ \ \ o-----------------------@ o--------\----------\---o o-----------------------o | |\ | \ \ | |```````````````````````| | | \ | \ @ | |```````````````````````| | | \| \ | |```````````````````````| | o--o o--o | \ o--o \o--o | |``````o--o```o--o``````| | / \ /````\ | |\ / \ /\ \ | |`````/ \`/ \`````| | / o``````\ | | \ / o @ \ | |````/ o \````| | / du / \``dv``\ | | \/ du /`\ dv \ | |```/ du / \ dv \```| | o / \``````o | | o\ /```\ o | |``o / \ o``| | | o o`````| | | | \ o`````o | | |``| o o |``| | | | |`````| | | | @ |``@--|-----|------@``| | | |``| | | o o`````| | | | o`````o | | |``| o o |``| | o \ /``````o | | o \```/ o | |``o \ / o``| | \ \ /``````/ | | \ \`/ / | |```\ \ / /```| | \ o``````/ | | \ o / | |````\ o /````| | \ / \````/ | | \ / \ / | |`````\ /`\ /`````| | o--o o--o | | o--o o--o | |``````o--o```o--o``````| | | | | |```````````````````````| | | | | |```````````````````````| | | | | |```````````````````````| o-----------------------o o-----------------------o o-----------------------o \ / \ / \ / \ EJ / \ J / \ EJ / \ / \ / \ / \ / o----------\---------/----------o \ / \ / | \ / | \ / \ / | \ / | \ / \ / | o-----o-----o | \ / \ / | /`````````````\ | \ / \ / | /```````````````\ | \ / o------\---/------o | /`````````````````\ | o------\---/------o | \ / | | /```````````````````\ | | \ / | | o--o--o | | /`````````````````````\ | | o--o--o | | /```````\ | | o```````````````````````o | | /```````\ | | /`````````\ | | |```````````````````````| | | /`````````\ | | o```````````o | | |```````````````````````| | | o```````````o | | |````dx`````| @----@ |```````````x`````@-----|------@ |``` dx ````| | | o```````````o | | |```````````````````````| | | o```````````o | | \`````````/ | | |```````````````````````| | | \`````````/ | | \```````/ | | o```````````````````````o | | \```````/ | | o-----o | | \`````````````````````/ | | o-----o | | | | \```````````````````/ | | | o-----------------o | \`````````````````/ | o-----------------o | \```````````````/ | | \`````````````/ | | o-----------o | | | | | o-------------------------------o Figure 56-b2. Secant Map of the Conjunction J = uv o-----------------------o | | | | | | | o--o o--o | | / \ / \ | | / o \ | | / du /`\ dv \ | | o /```\ o | | | o`````o | | | | |`````| | | | | o`````o | | | o \```/ o | | \ \`/ / | | \ o / | | \ / \ / | | o--o o--o | | | | | | | o-----------------------@ \ o-----------------------o \ | | \ | | \ | | \ | o--o o--o | \ | /````\ / \ | \ | /``````o \ | \ | /``du``/ \ dv \ | \ | o``````/ \ o | \ | |`````o o | @ \ | |`````| | | |\ \ | |`````o o | | \ \ | o``````\ / o | \ \ | \``````\ / / | \ \ | \``````o / | \ \ | \````/ \ / | \ \ | o--o o--o | \ \ | | \ \ | | \ \ | | \ \ o-----------------------o \ \ \ \ o-----------------------@ o--------\----------\---o o-----------------------o | |\ | \ \ | | | | | \ | \ @ | | | | | \| \ | | | | o--o o--o | \ o--o \o--o | | o--o o--o | | / \ /````\ | |\ / \ /\ \ | | /````\ /````\ | | / o``````\ | | \ / o @ \ | | /``````o``````\ | | / du / \``dv``\ | | \/ du /`\ dv \ | | /``du``/`\``dv``\ | | o / \``````o | | o\ /```\ o | | o``````/```\``````o | | | o o`````| | | | \ o`````o | | | |`````o`````o`````| | | | | |`````| | | | @ |``@--|-----|------@ |`````|`````|`````| | | | o o`````| | | | o`````o | | | |`````o`````o`````| | | o \ /``````o | | o \```/ o | | o``````\```/``````o | | \ \ /``````/ | | \ \`/ / | | \``````\`/``````/ | | \ o``````/ | | \ o / | | \``````o``````/ | | \ / \````/ | | \ / \ / | | \````/ \````/ | | o--o o--o | | o--o o--o | | o--o o--o | | | | | | | | | | | | | | | | | | | o-----------------------o o-----------------------o o-----------------------o \ / \ / \ / \ DJ / \ J / \ DJ / \ / \ / \ / \ / o----------\---------/----------o \ / \ / | \ / | \ / \ / | \ / | \ / \ / | o-----o-----o | \ / \ / | /`````````````\ | \ / \ / | /```````````````\ | \ / o------\---/------o | /`````````````````\ | o------\---/------o | \ / | | /```````````````````\ | | \ / | | o--o--o | | /`````````````````````\ | | o--o--o | | /```````\ | | o```````````````````````o | | /```````\ | | /`````````\ | | |```````````````````````| | | /`````````\ | | o```````````o | | |```````````````````````| | | o```````````o | | |````dx`````| @----@ |```````````x`````@-----|------@ |``` dx ````| | | o```````````o | | |```````````````````````| | | o```````````o | | \`````````/ | | |```````````````````````| | | \`````````/ | | \```````/ | | o```````````````````````o | | \```````/ | | o-----o | | \`````````````````````/ | | o-----o | | | | \```````````````````/ | | | o-----------------o | \`````````````````/ | o-----------------o | \```````````````/ | | \`````````````/ | | o-----------o | | | | | o-------------------------------o Figure 56-b3. Chord Map of the Conjunction J = uv o-----------------------o | | | | | | | o--o o--o | | / \ / \ | | / o \ | | / du / \ dv \ | | o / \ o | | | o o | | | | | | | | | | o o | | | o \ / o | | \ \ / / | | \ o / | | \ / \ / | | o--o o--o | | | | | | | o-----------------------@ \ o-----------------------o \ | | \ | | \ | | \ | o--o o--o | \ | /````\ / \ | \ | /``````o \ | \ | /``du``/`\ dv \ | \ | o``````/```\ o | \ | |`````o`````o | @ \ | |`````|`````| | |\ \ | |`````o`````o | | \ \ | o``````\```/ o | \ \ | \``````\`/ / | \ \ | \``````o / | \ \ | \````/ \ / | \ \ | o--o o--o | \ \ | | \ \ | | \ \ | | \ \ o-----------------------o \ \ \ \ o-----------------------@ o--------\----------\---o o-----------------------o | |\ | \ \ | | | | | \ | \ @ | | | | | \| \ | | | | o--o o--o | \ o--o \o--o | | o--o o--o | | / \ /````\ | |\ / \ /\ \ | | /````\ /````\ | | / o``````\ | | \ / o @ \ | | /``````o``````\ | | / du /`\``dv``\ | | \/ du /`\ dv \ | | /``du``/ \``dv``\ | | o /```\``````o | | o\ /```\ o | | o``````/ \``````o | | | o`````o`````| | | | \ o`````o | | | |`````o o`````| | | | |`````|`````| | | | @ |``@--|-----|------@ |`````| |`````| | | | o`````o`````| | | | o`````o | | | |`````o o`````| | | o \```/``````o | | o \```/ o | | o``````\ /``````o | | \ \`/``````/ | | \ \`/ / | | \``````\ /``````/ | | \ o``````/ | | \ o / | | \``````o``````/ | | \ / \````/ | | \ / \ / | | \````/ \````/ | | o--o o--o | | o--o o--o | | o--o o--o | | | | | | | | | | | | | | | | | | | o-----------------------o o-----------------------o o-----------------------o \ / \ / \ / \ dJ / \ J / \ dJ / \ / \ / \ / \ / o----------\---------/----------o \ / \ / | \ / | \ / \ / | \ / | \ / \ / | o-----o-----o | \ / \ / | /`````````````\ | \ / \ / | /```````````````\ | \ / o------\---/------o | /`````````````````\ | o------\---/------o | \ / | | /```````````````````\ | | \ / | | o--o--o | | /`````````````````````\ | | o--o--o | | /```````\ | | o```````````````````````o | | /```````\ | | /`````````\ | | |```````````````````````| | | /`````````\ | | o```````````o | | |```````````````````````| | | o```````````o | | |````dx`````| @----@ |```````````x`````@-----|------@ |``` dx ````| | | o```````````o | | |```````````````````````| | | o```````````o | | \`````````/ | | |```````````````````````| | | \`````````/ | | \```````/ | | o```````````````````````o | | \```````/ | | o-----o | | \`````````````````````/ | | o-----o | | | | \```````````````````/ | | | o-----------------o | \`````````````````/ | o-----------------o | \```````````````/ | | \`````````````/ | | o-----------o | | | | | o-------------------------------o Figure 56-b4. Tangent Map of the Conjunction J = uv o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D70 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Operator Diagrams for the Conjunction J = uv o o //\ /X\ ////\ /XXX\ //////\ oXXXXXo ////////\ /X\XXX/X\ //////////\ /XXX\X/XXX\ o///////////o oXXXXXoXXXXXo / \////////// \ / \XXX/X\XXX/ \ / \//////// \ / \X/XXX\X/ \ / \////// \ o oXXXXXo o / \//// \ / \ / \XXX/ \ / \ / \// \ / \ / \X/ \ / \ o o o o o o o o |\ / \ /| |\ / \ / \ / \ /| | \ / \ / | | \ / \ / \ / \ / | | \ / \ / | | o o o o | | \ / \ / | | |\ / \ / \ /| | | u \ / \ / v | |u | \ / \ / \ / | v| o-----o o-----o o--+--o o o--+--o \ / | \ / \ / | \ / | du \ / \ / dv | \ / o-----o o-----o \ / \ / \ / \ / o o U% $e$ $E$U% o------------------>o | | | | | | | | J | | $e$J | | | | | | v v o------------------>o X% $e$ $E$X% o o //\ /X\ ////\ /XXX\ //////\ /XXXXX\ ////////\ /XXXXXXX\ //////////\ /XXXXXXXXX\ ////////////o oXXXXXXXXXXXo ///////////// \ //\XXXXXXXXX/\\ ///////////// \ ////\XXXXXXX/\\\\ ///////////// \ //////\XXXXX/\\\\\\ ///////////// \ ////////\XXX/\\\\\\\\ ///////////// \ //////////\X/\\\\\\\\\\ o//////////// o o///////////o\\\\\\\\\\\o |\////////// / |\////////// \\\\\\\\\\/| | \//////// / | \//////// \\\\\\\\/ | | \////// / | \////// \\\\\\/ | | \//// / | \//// \\\\/ | | x \// / | x \// \\/ dx | o-----o / o-----o o-----o \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / o o Figure 57-1. Radius Operator Diagram for the Conjunction J = uv o o //\ /X\ ////\ /XXX\ //////\ oXXXXXo ////////\ //\XXX//\ //////////\ ////\X////\ o///////////o o/////o/////o / \////////// \ /\\/////\////\\ / \//////// \ /\\\\/////\//\\\\ / \////// \ o\\\\\o/////o\\\\\o / \//// \ / \\\\/ \//// \\\\/ \ / \// \ / \\/ \// \\/ \ o o o o o o o o |\ / \ /| |\ / \ /\\ / \ /| | \ / \ / | | \ / \ /\\\\ / \ / | | \ / \ / | | o o\\\\\o o | | \ / \ / | | |\ / \\\\/ \ /| | | u \ / \ / v | |u | \ / \\/ \ / | v| o-----o o-----o o--+--o o o--+--o \ / | \ / \ / | \ / | du \ / \ / dv | \ / o-----o o-----o \ / \ / \ / \ / o o U% $E$ $E$U% o------------------>o | | | | | | | | J | | $E$J | | | | | | v v o------------------>o X% $E$ $E$X% o o //\ /X\ ////\ /XXX\ //////\ /XXXXX\ ////////\ /XXXXXXX\ //////////\ /XXXXXXXXX\ ////////////o oXXXXXXXXXXXo ///////////// \ //\XXXXXXXXX/\\ ///////////// \ ////\XXXXXXX/\\\\ ///////////// \ //////\XXXXX/\\\\\\ ///////////// \ ////////\XXX/\\\\\\\\ ///////////// \ //////////\X/\\\\\\\\\\ o//////////// o o///////////o\\\\\\\\\\\o |\////////// / |\////////// \\\\\\\\\\/| | \//////// / | \//////// \\\\\\\\/ | | \////// / | \////// \\\\\\/ | | \//// / | \//// \\\\/ | | x \// / | x \// \\/ dx | o-----o / o-----o o-----o \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / o o Figure 57-2. Secant Operator Diagram for the Conjunction J = uv o o //\ //\ ////\ ////\ //////\ o/////o ////////\ /X\////X\ //////////\ /XXX\//XXX\ o///////////o oXXXXXoXXXXXo / \////////// \ /\\XXX/X\XXX/\\ / \//////// \ /\\\\X/XXX\X/\\\\ / \////// \ o\\\\\oXXXXXo\\\\\o / \//// \ / \\\\/ \XXX/ \\\\/ \ / \// \ / \\/ \X/ \\/ \ o o o o o o o o |\ / \ /| |\ / \ /\\ / \ /| | \ / \ / | | \ / \ /\\\\ / \ / | | \ / \ / | | o o\\\\\o o | | \ / \ / | | |\ / \\\\/ \ /| | | u \ / \ / v | |u | \ / \\/ \ / | v| o-----o o-----o o--+--o o o--+--o \ / | \ / \ / | \ / | du \ / \ / dv | \ / o-----o o-----o \ / \ / \ / \ / o o U% $D$ $E$U% o------------------>o | | | | | | | | J | | $D$J | | | | | | v v o------------------>o X% $D$ $E$X% o o //\ /X\ ////\ /XXX\ //////\ /XXXXX\ ////////\ /XXXXXXX\ //////////\ /XXXXXXXXX\ ////////////o oXXXXXXXXXXXo ///////////// \ //\XXXXXXXXX/\\ ///////////// \ ////\XXXXXXX/\\\\ ///////////// \ //////\XXXXX/\\\\\\ ///////////// \ ////////\XXX/\\\\\\\\ ///////////// \ //////////\X/\\\\\\\\\\ o//////////// o o///////////o\\\\\\\\\\\o |\////////// / |\////////// \\\\\\\\\\/| | \//////// / | \//////// \\\\\\\\/ | | \////// / | \////// \\\\\\/ | | \//// / | \//// \\\\/ | | x \// / | x \// \\/ dx | o-----o / o-----o o-----o \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / o o Figure 57-3. Chord Operator Diagram for the Conjunction J = uv o o //\ //\ ////\ ////\ //////\ o/////o ////////\ /X\////X\ //////////\ /XXX\//XXX\ o///////////o oXXXXXoXXXXXo / \////////// \ /\\XXX//\XXX/\\ / \//////// \ /\\\\X////\X/\\\\ / \////// \ o\\\\\o/////o\\\\\o / \//// \ / \\\\/\\////\\\\\/ \ / \// \ / \\/\\\\//\\\\\/ \ o o o o o\\\\\o\\\\\o o |\ / \ /| |\ / \\\\/ \\\\/ \ /| | \ / \ / | | \ / \\/ \\/ \ / | | \ / \ / | | o o o o | | \ / \ / | | |\ / \ / \ /| | | u \ / \ / v | |u | \ / \ / \ / | v| o-----o o-----o o--+--o o o--+--o \ / | \ / \ / | \ / | du \ / \ / dv | \ / o-----o o-----o \ / \ / \ / \ / o o U% $T$ $E$U% o------------------>o | | | | | | | | J | | $T$J | | | | | | v v o------------------>o X% $T$ $E$X% o o //\ /X\ ////\ /XXX\ //////\ /XXXXX\ ////////\ /XXXXXXX\ //////////\ /XXXXXXXXX\ ////////////o oXXXXXXXXXXXo ///////////// \ //\XXXXXXXXX/\\ ///////////// \ ////\XXXXXXX/\\\\ ///////////// \ //////\XXXXX/\\\\\\ ///////////// \ ////////\XXX/\\\\\\\\ ///////////// \ //////////\X/\\\\\\\\\\ o//////////// o o///////////o\\\\\\\\\\\o |\////////// / |\////////// \\\\\\\\\\/| | \//////// / | \//////// \\\\\\\\/ | | \////// / | \////// \\\\\\/ | | \//// / | \//// \\\\/ | | x \// / | x \// \\/ dx | o-----o / o-----o o-----o \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / o o Figure 57-4. Tangent Functor Diagram for the Conjunction J = uv o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D71 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | The past and present wilt . . . . I have filled them and | emptied them, | And proceed to fill my next fold of the future. | | Walt Whitman, 'Leaves of Grass', [Whi, 87] Taking Aim at Higher Dimensional Targets In the next Subdivision I consider a logical transformation F that has the concrete type F : [u, v] -> [x, y] and the abstract type F : [B^2] -> [B^2]. From the standpoint of propositional calculus, the task of understanding such a transformation is naturally approached by parsing it into component maps with 1-dimensional ranges, as follows: o-----------------------------------------------------------o | | | F = <f, g> = <F_1, F_2> : [u, v] -> [x, y] | | | | where f = F_1 : [u, v] -> [x] | | | | and g = F_2 : [u, v] -> [y] | | | o-----------------------------------------------------------o Then one tackles the separate components, now viewed as propositions F_i : U -> B, one at a time. At the completion of this analytic phase, one returns to the task of synthesizing all of these partial and transient impressions into an agile form of integrity, a solidly coordinated and deeply integrated comprehension of the ongoing transformation. (Very often, of course, in tangling with refractory cases, one never gets as far as the beginning again.) o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D72 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Taking Aim at Higher Dimensional Targets (concl.) Let us now refer to the dimension of the target space or codomain as the "toll" (or "tole") of a transformation, as distinguished from the dimension of the range or image that is customarily called the "rank". When we keep to transformations with a toll of 1, as J : [u, v] -> [x], we tend to get lazy about distinguishing a logical transformation from its component propositions. However, if we deal with transformations of a higher toll, this form of indolence can no longer be tolerated. Well, perhaps we can carry it a little further. After all, the operator result WJ : EU% -> EX% is a map of toll 2, and cannot be unfolded in one piece as a proposition. But when a map has rank 1, like !e!J : EU -> X c EX or dJ : EU -> dX c EX, we naturally choose to concentrate on the 1-dimensional range of the operator result WJ, ignoring the final difference in quality between the spaces X and dX, and view WJ as a proposition about EU. In this way, an initial ambivalence about the role of the operand J conveys a double duty to the result WJ. The pivot that is formed by our focus of attention is essential to the linkage that transfers this double moment, as the whole process takes its bearing and wheels around the precise measure of a narrow bead that we can draw on the range of WJ. This is the escapement that it takes to get away with what may otherwise seem to be a simple duplicity, and this is the tolerance that is needed to counterbalance a certain arrogance of equivocation, by all of which machinations we make ourselves free to indicate the operator results WJ as propositions or as transformations, indifferently. But that's it, and no further. Neglect of these distinctions in range and target universes of higher dimensions is bound to cause a hopeless confusion. To guard against these adverse prospects, Tables 58 and 59 lay the groundwork for discussing a typical map F : [B^2] -> [B^2], and begin to pave the way, to some extent, for discussing any transformation of the form F : [B^n] -> [B^k]. Table 58. Cast of Characters: Expansive Subtypes of Objects and Operators o------o-------------------------o------------------o----------------------------o | Item | Notation | Description | Type | o------o-------------------------o------------------o----------------------------o | | | | | | U% | = [u, v] | Source Universe | [B^n] | | | | | | o------o-------------------------o------------------o----------------------------o | | | | | | X% | = [x, y] | Target Universe | [B^k] | | | = [f, g] | | | | | | | | o------o-------------------------o------------------o----------------------------o | | | | | | EU% | = [u, v, du, dv] | Extended | [B^n x D^n] | | | | Source Universe | | | | | | | o------o-------------------------o------------------o----------------------------o | | | | | | EX% | = [x, y, dx, dy] | Extended | [B^k x D^k] | | | = [f, g, df, dg] | Target Universe | | | | | | | o------o-------------------------o------------------o----------------------------o | | | | | | F | F = <f, g> : U% -> X% | Transformation, | [B^n] -> [B^k] | | | | or Mapping | | | | | | | o------o-------------------------o------------------o----------------------------o | | | | | | | f, g : U -> B | Proposition, | B^n -> B | | | | special case | | | f | f : U -> [x] c X% | of a mapping, | c (B^n, B^n -> B) | | | | or component | | | g | g : U -> [y] c X% | of a mapping. | = (B^n +-> B) = [B^n] | | | | | | o------o-------------------------o------------------o----------------------------o | | | | | | W | W : | Operator | | | | U% -> EU%, | | [B^n] -> [B^n x D^n], | | | X% -> EX%, | | [B^k] -> [B^k x D^k], | | | (U%->X%)->(EU%->EX%), | | ([B^n] -> [B^k]) | | | for each W among: | | -> | | | !e!, !h!, E, D, d | | ([B^n x D^n]->[B^k x D^k]) | | | | | | o------o-------------------------o------------------o----------------------------o | | | | | !e! | | Tacit Extension Operator !e! | | !h! | | Trope Extension Operator !h! | | E | | Enlargement Operator E | | D | | Difference Operator D | | d | | Differential Operator d | | | | | o------o-------------------------o------------------o----------------------------o | | | | | | $W$ | $W$ : | Operator | | | | U% -> $T$U% = EU%, | | [B^n] -> [B^n x D^n], | | | X% -> $T$X% = EX%, | | [B^k] -> [B^k x D^k], | | | (U%->X%)->($T$U%->$T$X%)| | ([B^n] -> [B^k]) | | | for each $W$ among: | | -> | | | $e$, $E$, $D$, $T$ | | ([B^n x D^n]->[B^k x D^k]) | | | | | | o------o-------------------------o------------------o----------------------------o | | | | | $e$ | | Radius Operator $e$ = <!e!, !h!> | | $E$ | | Secant Operator $E$ = <!e!, E > | | $D$ | | Chord Operator $D$ = <!e!, D > | | $T$ | | Tangent Functor $T$ = <!e!, d > | | | | | o------o-------------------------o-----------------------------------------------o Table 59. Synopsis of Terminology: Restrictive and Alternative Subtypes o--------------o----------------------o--------------------o----------------------o | | Operator | Proposition | Transformation | | | or | or | or | | | Operand | Component | Mapping | o--------------o----------------------o--------------------o----------------------o | | | | | | Operand | F = <F_1, F_2> | F_i : <|u,v|> -> B | F : [u, v] -> [x, y] | | | | | | | | F = <f, g> : U -> X | F_i : B^n -> B | F : B^n -> B^k | | | | | | o--------------o----------------------o--------------------o----------------------o | | | | | | Tacit | !e! : | !e!F_i : | !e!F : | | Extension | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> B | [u,v,du,dv]->[x, y] | | | (U%->X%)->(EU%->X%) | B^n x D^n -> B | [B^n x D^n]->[B^k] | | | | | | o--------------o----------------------o--------------------o----------------------o | | | | | | Trope | !h! : | !h!F_i : | !h!F : | | Extension | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] | | | (U%->X%)->(EU%->dX%) | B^n x D^n -> D | [B^n x D^n]->[D^k] | | | | | | o--------------o----------------------o--------------------o----------------------o | | | | | | Enlargement | E : | EF_i : | EF : | | Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] | | | (U%->X%)->(EU%->dX%) | B^n x D^n -> D | [B^n x D^n]->[D^k] | | | | | | o--------------o----------------------o--------------------o----------------------o | | | | | | Difference | D : | DF_i : | DF : | | Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] | | | (U%->X%)->(EU%->dX%) | B^n x D^n -> D | [B^n x D^n]->[D^k] | | | | | | o--------------o----------------------o--------------------o----------------------o | | | | | | Differential | d : | dF_i : | dF : | | Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] | | | (U%->X%)->(EU%->dX%) | B^n x D^n -> D | [B^n x D^n]->[D^k] | | | | | | o--------------o----------------------o--------------------o----------------------o | | | | | | Remainder | r : | rF_i : | rF : | | Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] | | | (U%->X%)->(EU%->dX%) | B^n x D^n -> D | [B^n x D^n]->[D^k] | | | | | | o--------------o----------------------o--------------------o----------------------o | | | | | | Radius | $e$ = <!e!, !h!> : | | $e$F : | | Operator | | | | | | U%->EU%, X%->EX%, | | [u, v, du, dv] -> | | | (U%->X%)->(EU%->EX%) | | [x, y, dx, dy], | | | | | | | | | | [B^n x D^n] -> | | | | | [B^k x D^k] | | | | | | o--------------o----------------------o--------------------o----------------------o | | | | | | Secant | $E$ = <!e!, E> : | | $E$F : | | Operator | | | | | | U%->EU%, X%->EX%, | | [u, v, du, dv] -> | | | (U%->X%)->(EU%->EX%) | | [x, y, dx, dy], | | | | | | | | | | [B^n x D^n] -> | | | | | [B^k x D^k] | | | | | | o--------------o----------------------o--------------------o----------------------o | | | | | | Chord | $D$ = <!e!, D> : | | $D$F : | | Operator | | | | | | U%->EU%, X%->EX%, | | [u, v, du, dv] -> | | | (U%->X%)->(EU%->EX%) | | [x, y, dx, dy], | | | | | | | | | | [B^n x D^n] -> | | | | | [B^k x D^k] | | | | | | o--------------o----------------------o--------------------o----------------------o | | | | | | Tangent | $T$ = <!e!, d> : | dF_i : | $T$F : | | Functor | | | | | | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u, v, du, dv] -> | | | (U%->X%)->(EU%->EX%) | | [x, y, dx, dy], | | | | | | | | | B^n x D^n -> D | [B^n x D^n] -> | | | | | [B^k x D^k] | | | | | | o--------------o----------------------o--------------------o----------------------o o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D73 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Transformations of Type B^2 -> B^2 To take up a slightly more complex example, but one that remains simple enough to pursue through a complete series of developments, consider the transformation from U% = [u, v] to X% = [x, y] that is defined by the following system of equations: o-----------------------------------------------------------o | | | x = f(u, v) = ((u)(v)) | | | | y = g(u, v) = ((u, v)) | | | o-----------------------------------------------------------o The component notation F = <F_1, F_2> = <f, g> : U% -> X% allows us to give a name and a type to this transformation, and permits us to define it by means of the compact description that follows: o-----------------------------------------------------------o | | | <x, y> = F<u, v> = <((u)(v)), ((u, v))> | | | o-----------------------------------------------------------o The information that defines the logical transformation F can be represented in the form of a truth table, as in Table 60. To cut down on subscripts in this example I continue to use plain letter equivalents for all components of spaces and maps. Table 60. Propositional Transformation o-------------o-------------o-------------o-------------o | u | v | f | g | o-------------o-------------o-------------o-------------o | | | | | | 0 | 0 | 0 | 1 | | | | | | | 0 | 1 | 1 | 0 | | | | | | | 1 | 0 | 1 | 0 | | | | | | | 1 | 1 | 1 | 1 | | | | | | o-------------o-------------o-------------o-------------o | | | ((u)(v)) | ((u, v)) | o-------------o-------------o-------------o-------------o o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D74 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Transformations of Type B^2 -> B^2 (cont.) Figure 61 shows how one might paint a picture of the logical transformation F on the canvass that was earlier primed for this purpose (way back in Figure 30). o-----------------------------------------------------o | U | | | | o-----------o o-----------o | | / \ / \ | | / o \ | | / / \ \ | | / / \ \ | | o o o o | | | | | | | | | u | | v | | | | | | | | | o o o o | | \ \ / / | | \ \ / / | | \ o / | | \ / \ / | | o-----------o o-----------o | | | | | o-----------------------------------------------------o / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ o-------------------------o o-------------------------o | U | |\U \\\\\\\\\\\\\\\\\\\\\\| | o---o o---o | |\\\\\\o---o\\\o---o\\\\\\| | //////\ //////\ | |\\\\\/ \\/ \\\\\\| | ////////o///////\ | |\\\\/ o \\\\\| | //////////\///////\ | |\\\/ /\\ \\\\| | o///////o///o///////o | |\\o o\\\o o\\| | |// u //|///|// v //| | |\\| u |\\\| v |\\| | o///////o///o///////o | |\\o o\\\o o\\| | \///////\////////// | |\\\\ \\/ /\\\| | \///////o//////// | |\\\\\ o /\\\\| | \////// \////// | |\\\\\\ /\\ /\\\\\| | o---o o---o | |\\\\\\o---o\\\o---o\\\\\\| | | |\\\\\\\\\\\\\\\\\\\\\\\\\| o-------------------------o o-------------------------o \ | | / \ | | / \ | | / \ f | | g / \ | | / \ | | / \ | | / \ | | / \ | | / \ | | / o-------\----|---------------------------|----/-------o | X \ | | / | | \| |/ | | o-----------o o-----------o | | //////////////\ /\\\\\\\\\\\\\\ | | ////////////////o\\\\\\\\\\\\\\\\ | | /////////////////X\\\\\\\\\\\\\\\\\ | | /////////////////XXX\\\\\\\\\\\\\\\\\ | | o///////////////oXXXXXo\\\\\\\\\\\\\\\o | | |///////////////|XXXXX|\\\\\\\\\\\\\\\| | | |////// x //////|XXXXX|\\\\\\ y \\\\\\| | | |///////////////|XXXXX|\\\\\\\\\\\\\\\| | | o///////////////oXXXXXo\\\\\\\\\\\\\\\o | | \///////////////\XXX/\\\\\\\\\\\\\\\/ | | \///////////////\X/\\\\\\\\\\\\\\\/ | | \///////////////o\\\\\\\\\\\\\\\/ | | \////////////// \\\\\\\\\\\\\\/ | | o-----------o o-----------o | | | | | o-----------------------------------------------------o Figure 61. Propositional Transformation o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D75 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Transformations of Type B^2 -> B^2 (cont.) Figure 62 extracts the gist of Figure 61, exemplifying a style of diagram that is adequate for most purposes. o-------------------------o o-------------------------o | U | |\U \\\\\\\\\\\\\\\\\\\\\\| | o---o o---o | |\\\\\\o---o\\\o---o\\\\\\| | //////\ //////\ | |\\\\\/ \\/ \\\\\\| | ////////o///////\ | |\\\\/ o \\\\\| | //////////\///////\ | |\\\/ /\\ \\\\| | o///////o///o///////o | |\\o o\\\o o\\| | |// u //|///|// v //| | |\\| u |\\\| v |\\| | o///////o///o///////o | |\\o o\\\o o\\| | \///////\////////// | |\\\\ \\/ /\\\| | \///////o//////// | |\\\\\ o /\\\\| | \////// \////// | |\\\\\\ /\\ /\\\\\| | o---o o---o | |\\\\\\o---o\\\o---o\\\\\\| | | |\\\\\\\\\\\\\\\\\\\\\\\\\| o-------------------------o o-------------------------o \ / \ / \ / \ / \ / \ / \ f / \ g / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / o---------\-----/---------------------\-----/---------o | X \ / \ / | | \ / \ / | | o-----------o o-----------o | | //////////////\ /\\\\\\\\\\\\\\ | | ////////////////o\\\\\\\\\\\\\\\\ | | /////////////////X\\\\\\\\\\\\\\\\\ | | /////////////////XXX\\\\\\\\\\\\\\\\\ | | o///////////////oXXXXXo\\\\\\\\\\\\\\\o | | |///////////////|XXXXX|\\\\\\\\\\\\\\\| | | |////// x //////|XXXXX|\\\\\\ y \\\\\\| | | |///////////////|XXXXX|\\\\\\\\\\\\\\\| | | o///////////////oXXXXXo\\\\\\\\\\\\\\\o | | \///////////////\XXX/\\\\\\\\\\\\\\\/ | | \///////////////\X/\\\\\\\\\\\\\\\/ | | \///////////////o\\\\\\\\\\\\\\\/ | | \////////////// \\\\\\\\\\\\\\/ | | o-----------o o-----------o | | | | | o-----------------------------------------------------o Figure 62. Propositional Transformation (Short Form) o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D76 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Transformations of Type B^2 -> B^2 (cont.) Figure 63 give a more complete picture of the transformation F, showing how the points of U% are transformed into points of X%. The lines that cross from one universe to the other trace the action that F induces on points, in other words, they depict the aspect of the transformation that acts as a mapping from points to points, and chart its effects on the elements that are variously called cells, points, positions, or singular propositions. o-----------------------------------------------------o |`U` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `| |` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `| |` ` ` ` ` ` o-----------o ` o-----------o ` ` ` ` ` `| |` ` ` ` ` `/' ' ' ' ' ' '\`/' ' ' ' ' ' '\` ` ` ` ` `| |` ` ` ` ` / ' ' ' ' ' ' ' o ' ' ' ' ' ' ' \ ` ` ` ` `| |` ` ` ` `/' ' ' ' ' ' ' '/^\' ' ' ' ' ' ' '\` ` ` ` `| |` ` ` ` / ' ' ' ' ' ' ' /^^^\ ' ' ' ' ' ' ' \ ` ` ` `| |` ` ` `o' ' ' ' ' ' ' 'o^^^^^o' ' ' ' ' ' ' 'o` ` ` `| |` ` ` `|' ' ' ' ' ' ' '|^^^^^|' ' ' ' ' ' ' '|` ` ` `| |` ` ` `|' ' ' ' u ' ' '|^^^^^|' ' ' v ' ' ' '|` ` ` `| |` ` ` `|' ' ' ' ' ' ' '|^^^^^|' ' ' ' ' ' ' '|` ` ` `| |` `@` `o' ' ' ' @ ' ' 'o^^@^^o' ' ' @ ' ' ' 'o` ` ` `| |` ` \ ` \ ' ' ' | ' ' ' \^|^/ ' ' ' | ' ' ' / ` ` ` `| |` ` `\` `\' ' ' | ' ' ' '\|/' ' ' ' | ' ' '/` ` ` ` `| |` ` ` \ ` \ ' ' | ' ' ' ' | ' ' ' ' | ' ' / ` ` ` ` `| |` ` ` `\` `\' ' | ' ' ' '/|\' ' ' ' | ' '/` ` ` ` ` `| |` ` ` ` \ ` o---|-------o | o-------|---o ` ` ` ` ` `| |` ` ` ` `\` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` `| |` ` ` ` ` \ ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` `| o-----------\----|---------|---------|----------------o " " \ | | | " " " " \ | | | " " " " \ | | | " " " " \| | | " " o-------------------------o \ | | o-------------------------o | U | |\ | | |`U```````````````````````| | o---o o---o | | \ | | |``````o---o```o---o``````| | /'''''\ /'''''\ | | \ | | |`````/ \`/ \`````| | /'''''''o'''''''\ | | \ | | |````/ o \````| | /'''''''/'\'''''''\ | | \ | | |```/ /`\ \```| | o'''''''o'''o'''''''o | | \ | | |``o o```o o``| | |'''u'''|'''|'''v'''| | | \ | | |``| u |```| v |``| | o'''''''o'''o'''''''o | | \ | | |``o o```o o``| | \'''''''\'/'''''''/ | | \| | |```\ \`/ /```| | \'''''''o'''''''/ | | \ | |````\ o /````| | \'''''/ \'''''/ | | |\ | |`````\ /`\ /`````| | o---o o---o | | | \ | |``````o---o```o---o``````| | | | | \ * |`````````````````````````| o-------------------------o | | \ / o-------------------------o \ | | | \ / | / \ ((u)(v)) | | | \/ | ((u, v)) / \ | | | /\ | / \ | | | / \ | / \ | | | / \ | / \ | | | / * | / \ | | | / | | / \ | | |/ | | / \ | | / | | / \ | | /| | | / o-------\----|---|-------/-|---------|---|----/-------o | X \ | | / | | | / | | \| | / | | |/ | | o---|----/--o | o-------|---o | | /' ' | ' / ' '\|/` ` ` ` | ` `\ | | / ' ' | '/' ' ' | ` ` ` ` | ` ` \ | | /' ' ' | / ' ' '/|\` ` ` ` | ` ` `\ | | / ' ' ' |/' ' ' /^|^\ ` ` ` | ` ` ` \ | | @ o' ' ' ' @ ' ' 'o^^@^^o` ` ` @ ` ` ` `o | | |' ' ' ' ' ' ' '|^^^^^|` ` ` ` ` ` ` `| | | |' ' ' ' f ' ' '|^^^^^|` ` ` g ` ` ` `| | | |' ' ' ' ' ' ' '|^^^^^|` ` ` ` ` ` ` `| | | o' ' ' ' ' ' ' 'o^^^^^o` ` ` ` ` ` ` `o | | \ ' ' ' ' ' ' ' \^^^/ ` ` ` ` ` ` ` / | | \' ' ' ' ' ' ' '\^/` ` ` ` ` ` ` `/ | | \ ' ' ' ' ' ' ' o ` ` ` ` ` ` ` / | | \' ' ' ' ' ' '/ \` ` ` ` ` ` `/ | | o-----------o o-----------o | | | | | o-----------------------------------------------------o Figure 63. Transformation of Positions o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D77 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Transformations of Type B^2 -> B^2 (cont.) Table 64 shows how the action of the transformation F on cells or points is computed in terms of coordinates. Table 64. Transformation of Positions o-----o----------o----------o-------o-------o--------o--------o-------------o | u v | x | y | x y | x(y) | (x)y | (x)(y) | X% = [x, y] | o-----o----------o----------o-------o-------o--------o--------o-------------o | | | | | | | | ^ | | 0 0 | 0 | 1 | 0 | 0 | 1 | 0 | | | | | | | | | | | | | 0 1 | 1 | 0 | 0 | 1 | 0 | 0 | F | | | | | | | | | = | | 1 0 | 1 | 0 | 0 | 1 | 0 | 0 | <f , g> | | | | | | | | | | | 1 1 | 1 | 1 | 1 | 0 | 0 | 0 | ^ | | | | | | | | | | | o-----o----------o----------o-------o-------o--------o--------o-------------o | | ((u)(v)) | ((u, v)) | u v | (u,v) | (u)(v) | 0 | U% = [u, v] | o-----o----------o----------o-------o-------o--------o--------o-------------o Table 65 extends this scheme from single cells to arbitrary regions of the source and target universes, and illustrates a form of computation that can be used to determine how a logical transformation acts on all of the propositions in the universe. The way that a transformation of positions affects the propositions, or any other structure that can be built on top of the positions, is normally called the "induced action" of the given transformation on the system of structures in question. Table 65. Induced Transformation on Propositions o------------o---------------------------------o------------o | X% | <--- F = <f , g> <--- | U% | o------------o----------o-----------o----------o------------o | | u = | 1 1 0 0 | = u | | | | v = | 1 0 1 0 | = v | | | f_i <x, y> o----------o-----------o----------o f_j <u, v> | | | x = | 1 1 1 0 | = f<u,v> | | | | y = | 1 0 0 1 | = g<u,v> | | o------------o----------o-----------o----------o------------o | | | | | | | f_0 | () | 0 0 0 0 | () | f_0 | | | | | | | | f_1 | (x)(y) | 0 0 0 1 | () | f_0 | | | | | | | | f_2 | (x) y | 0 0 1 0 | (u)(v) | f_1 | | | | | | | | f_3 | (x) | 0 0 1 1 | (u)(v) | f_1 | | | | | | | | f_4 | x (y) | 0 1 0 0 | (u, v) | f_6 | | | | | | | | f_5 | (y) | 0 1 0 1 | (u, v) | f_6 | | | | | | | | f_6 | (x, y) | 0 1 1 0 | (u v) | f_7 | | | | | | | | f_7 | (x y) | 0 1 1 1 | (u v) | f_7 | | | | | | | o------------o----------o-----------o----------o------------o | | | | | | | f_8 | x y | 1 0 0 0 | u v | f_8 | | | | | | | | f_9 | ((x, y)) | 1 0 0 1 | u v | f_8 | | | | | | | | f_10 | y | 1 0 1 0 | ((u, v)) | f_9 | | | | | | | | f_11 | (x (y)) | 1 0 1 1 | ((u, v)) | f_9 | | | | | | | | f_12 | x | 1 1 0 0 | ((u)(v)) | f_14 | | | | | | | | f_13 | ((x) y) | 1 1 0 1 | ((u)(v)) | f_14 | | | | | | | | f_14 | ((x)(y)) | 1 1 1 0 | (()) | f_15 | | | | | | | | f_15 | (()) | 1 1 1 1 | (()) | f_15 | | | | | | | o------------o----------o-----------o----------o------------o o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D78 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Transformations of Type B^2 -> B^2 (cont.) Given the alphabets !U! = {u, v} and !X! = {x, y}, along with the corresponding universes of discourse U% and X% ~=~ [B^2], how many logical transformations of the general form G = <G_1, G_2> : U% -> X% are there? Since G_1 and G_2 can be any propositions of the type B^2 -> B, there are 2^4 = 16 choices for each of the maps G_1 and G_2, and thus there are 2^4 * 2^4 = 2^8 = 256 different mappings altogether of the form G : U% -> X%. The set of all functions of a given type is customarily denoted by placing its type indicator in parentheses, in the present instance writing (U% -> X%) = {G : U% -> X%}, and so the cardinality of this "function space" can be most conveniently summed up by writing |(U% -> X%)| = |(B^2 -> B^2)| = 4^4 = 256. Given any transformation G = <G_1, G_2> : U% -> X% of this type, one can define a couple of further transformations, related to G, that operate between the extended universes, EU% and EX%, of its source and target domains. First, the enlargement map (or the secant transformation) EG = <EG_1, EG_2> : EU% -> EX% is defined by the following set of component equations: o-------------------------------------------------o | | | EG_i = G_i <u + du, v + dv> | | | o-------------------------------------------------o Second, the difference map (or the chordal transformation) DG = <DG_1, DG_2> : EU% -> EX% is defined in component-wise fashion as the boolean sum of the initial proposition G_i and the enlarged proposition EG_i, for i = 1, 2, according to the following set of equations: o-------------------------------------------------o | | | DG_i = G_i <u, v> + EG_i <u, v, du, dv> | | | | = G_i <u, v> + G_i <u + du, v + dv> | | | o-------------------------------------------------o Maintaining a strict analogy with ordinary difference calculus would perhaps have us write DG_i = EG_i - G_i, but the sum and difference operations are the same thing in boolean arithmetic. It is more often natural in the logical context to consider an initial proposition q, then to compute the enlargement Eq, and finally to determine the difference Dq = q + Eq, so we let the variant order of terms reflect this sequence of considerations. Viewed in this light the difference operator D is imagined to be a function of very wide scope and polymorphic application, one that is able to realize the association between each transformation G and its difference map DG, for instance, taking the function space (U% -> X%) into (EU% -> EX%). Given the interpretive flexibility of contexts in which we are allowing a proposition to appear, it should be clear that an operator of this scope is not at all a trivial matter to define properly, and may take some trouble to work out. For the moment, let's content ourselves with returning to particular cases. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D79 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Transformations of Type B^2 -> B^2 (cont.) In their application to the present example, namely, the logical transformation F = <f, g> = <((u)(v)), ((u, v))>, the operators E and D respectively produce the enlarged map EF = <Ef, Eg> and the difference map DF = <Df, Dg>, whose components can be given as follows, if the reader, in lieu of a special font for the logical parentheses, can forgive a syntactically bilingual formulation: o-------------------------------------------------o | | | Ef = ((u + du)(v + dv)) | | | | Eg = ((u + du, v + dv)) | | | o-------------------------------------------------o o-------------------------------------------------o | | | Df = ((u)(v)) + ((u + du)(v + dv)) | | | | Dg = ((u, v)) + ((u + du, v + dv)) | | | o-------------------------------------------------o But these initial formulas are purely definitional, and help us little in understanding either the purpose of the operators or the meaning of their results. Working symbolically, let us apply the same method to the separate components f and g that we earlier used on J. This work is recorded in Appendix 1 and a summary of the results is presented in Tables 66-i and 66-ii. Table 66-i. Computation Summary for f<u, v> = ((u)(v)) o--------------------------------------------------------------------------------o | | | !e!f = uv. 1 + u(v). 1 + (u)v. 1 + (u)(v). 0 | | | | Ef = uv. (du dv) + u(v). (du (dv)) + (u)v.((du) dv) + (u)(v).((du)(dv)) | | | | Df = uv. du dv + u(v). du (dv) + (u)v. (du) dv + (u)(v).((du)(dv)) | | | | df = uv. 0 + u(v). du + (u)v. dv + (u)(v). (du, dv) | | | | rf = uv. du dv + u(v). du dv + (u)v. du dv + (u)(v). du dv | | | o--------------------------------------------------------------------------------o Table 66-ii. Computation Summary for g<u, v> = ((u, v)) o--------------------------------------------------------------------------------o | | | !e!g = uv. 1 + u(v). 0 + (u)v. 0 + (u)(v). 1 | | | | Eg = uv.((du, dv)) + u(v). (du, dv) + (u)v. (du, dv) + (u)(v).((du, dv)) | | | | Dg = uv. (du, dv) + u(v). (du, dv) + (u)v. (du, dv) + (u)(v). (du, dv) | | | | dg = uv. (du, dv) + u(v). (du, dv) + (u)v. (du, dv) + (u)(v). (du, dv) | | | | rg = uv. 0 + u(v). 0 + (u)v. 0 + (u)(v). 0 | | | o--------------------------------------------------------------------------------o o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D80 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Transformations of Type B^2 -> B^2 (cont.) Table 67 shows how to compute the analytic series for F = <f, g> = <((u)(v)), ((u, v))> in terms of coordinates, and Table 68 recaps these results in symbolic terms, agreeing with earlier derivations. Table 67. Computation of an Analytic Series in Terms of Coordinates o--------o-------o-------o--------o-------o-------o-------o-------o | u v | du dv | u' v' | f g | Ef Eg | Df Dg | df dg | rf rg | o--------o-------o-------o--------o-------o-------o-------o-------o | | | | | | | | | | 0 0 | 0 0 | 0 0 | 0 1 | 0 1 | 0 0 | 0 0 | 0 0 | | | | | | | | | | | | 0 1 | 0 1 | | 1 0 | 1 1 | 1 1 | 0 0 | | | | | | | | | | | | 1 0 | 1 0 | | 1 0 | 1 1 | 1 1 | 0 0 | | | | | | | | | | | | 1 1 | 1 1 | | 1 1 | 1 0 | 0 0 | 1 0 | | | | | | | | | | o--------o-------o-------o--------o-------o-------o-------o-------o | | | | | | | | | | 0 1 | 0 0 | 0 1 | 1 0 | 1 0 | 0 0 | 0 0 | 0 0 | | | | | | | | | | | | 0 1 | 0 0 | | 0 1 | 1 1 | 1 1 | 0 0 | | | | | | | | | | | | 1 0 | 1 1 | | 1 1 | 0 1 | 0 1 | 0 0 | | | | | | | | | | | | 1 1 | 1 0 | | 1 0 | 0 0 | 1 0 | 1 0 | | | | | | | | | | o--------o-------o-------o--------o-------o-------o-------o-------o | | | | | | | | | | 1 0 | 0 0 | 1 0 | 1 0 | 1 0 | 0 0 | 0 0 | 0 0 | | | | | | | | | | | | 0 1 | 1 1 | | 1 1 | 0 1 | 0 1 | 0 0 | | | | | | | | | | | | 1 0 | 0 0 | | 0 1 | 1 1 | 1 1 | 0 0 | | | | | | | | | | | | 1 1 | 0 1 | | 1 0 | 0 0 | 1 0 | 1 0 | | | | | | | | | | o--------o-------o-------o--------o-------o-------o-------o-------o | | | | | | | | | | 1 1 | 0 0 | 1 1 | 1 1 | 1 1 | 0 0 | 0 0 | 0 0 | | | | | | | | | | | | 0 1 | 1 0 | | 1 0 | 0 1 | 0 1 | 0 0 | | | | | | | | | | | | 1 0 | 0 1 | | 1 0 | 0 1 | 0 1 | 0 0 | | | | | | | | | | | | 1 1 | 0 0 | | 0 1 | 1 0 | 0 0 | 1 0 | | | | | | | | | | o--------o-------o-------o--------o-------o-------o-------o-------o Table 68. Computation of an Analytic Series in Symbolic Terms o-----o-----o------------o----------o----------o----------o----------o----------o | u v | f g | Df | Dg | df | dg | rf | rf | o-----o-----o------------o----------o----------o----------o----------o----------o | | | | | | | | | | 0 0 | 0 1 | ((du)(dv)) | (du, dv) | (du, dv) | (du, dv) | du dv | () | | | | | | | | | | | 0 1 | 1 0 | (du) dv | (du, dv) | dv | (du, dv) | du dv | () | | | | | | | | | | | 1 0 | 1 0 | du (dv) | (du, dv) | du | (du, dv) | du dv | () | | | | | | | | | | | 1 1 | 1 1 | du dv | (du, dv) | () | (du, dv) | du dv | () | | | | | | | | | | o-----o-----o------------o----------o----------o----------o----------o----------o o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D81 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Transformations of Type B^2 -> B^2 (cont.) Figure 69 gives a graphical picture of the difference map DF = <Df, Dg> for the transformation F = <f, g> = <((u)(v)), ((u, v))>. This depicts the same information about Df and Dg that was given in the corresponding rows of the computation summary in Tables 66-i and 66-ii, excerpted here: o-------------------------------------------------------------------------o | | | Df = uv. du dv + u(v). du (dv) + (u)v.(du) dv + (u)(v).((du)(dv)) | | | | Dg = uv.(du, dv) + u(v).(du, dv) + (u)v.(du, dv) + (u)(v). (du, dv) | | | o-------------------------------------------------------------------------o o-----------------------------------o o-----------------------------------o | U | |`U`````````````````````````````````| | | |```````````````````````````````````| | ^ | |```````````````````````````````````| | | | |```````````````````````````````````| | o-------o | o-------o | |```````o-------o```o-------o```````| | ^ /`````````\|/`````````\ ^ | | ^ ```/ ^ \`/ ^ \``` ^ | | \ /```````````|```````````\ / | |``\``/ \ o / \``/``| | \/`````u`````/|\`````v`````\/ | |```\/ u \/`\/ v \/```| | /\``````````/`|`\``````````/\ | |```/\ /\`/\ /\```| | o``\````````o``@``o````````/``o | |``o \ o``@``o / o``| | |```\```````|`````|```````/```| | |``| \ |`````| / |``| | |````@``````|`````|``````@````| | |``| @-------->`<--------@ |``| | |```````````|`````|```````````| | |``| |`````| |``| | o```````````o` ^ `o```````````o | |``o o`````o o``| | \```````````\`|`/```````````/ | |```\ \```/ /```| | \```` ^ ````\|/```` ^ ````/ | |````\ ^ \`/ ^ /````| | \`````\`````|`````/`````/ | |`````\ \ o / /`````| | \`````\```/|\```/`````/ | |``````\ \ /`\ / /``````| | o-----\-o | o-/-----o | |```````o-----\-o```o-/-----o```````| | \ | / | |``````````````\`````/``````````````| | \ | / | |```````````````\```/```````````````| | \|/ | |````````````````\`/````````````````| | @ | |`````````````````@`````````````````| o-----------------------------------o o-----------------------------------o \ / \ / \ / \ / \ ((u)(v)) / \ ((u, v)) / \ / \ / \ / \ / o----------\-------------/-----------------------\-------------/----------o | X \ / \ / | | \ / \ / | | \ / \ / | | o----------------o o----------------o | | / \ / \ | | / o \ | | / / \ \ | | / / \ \ | | / / \ \ | | / / \ \ | | / / \ \ | | o o o o | | | | | | | | | | | | | | | f | | g | | | | | | | | | | | | | | | o o o o | | \ \ / / | | \ \ / / | | \ \ / / | | \ \ / / | | \ \ / / | | \ o / | | \ / \ / | | o----------------o o----------------o | | | | | | | o-------------------------------------------------------------------------o Figure 69. Difference Map of F = <f, g> = <((u)(v)), ((u, v))> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D82 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Transformations of Type B^2 -> B^2 (cont.) Figure 70-a shows a graphical way of picturing the tangent functor map dF = <df, dg> for the transformation F = <f, g> = <((u)(v)), ((u, v))>. This amounts to the same information about df and dg that was given in the computation summary of Tables 66-i and 66-ii, the relevant rows of which are repeated here: o-------------------------------------------------------------------------------o | | | df = uv. 0 + u(v). du + (u)v. dv + (u)(v).(du, dv) | | | | dg = uv.(du, dv) + u(v).(du, dv) + (u)v.(du, dv) + (u)(v).(du, dv) | | | o-------------------------------------------------------------------------------o o o / \ / \ / \ / \ / \ / O \ / \ o /@\ o / \ / \ / \ / \ / \ / \ / O \ / O \ / O \ o /@\ o o /@\ o /@\ o / \ / \ / \ \ / \ \ / \ / \ / \ / \ / \ / \ / \ / \ / O \ / O \ / O \ / \ / \ o /@ o /@\ o /@ o / \ / \ / \ \ / \ / \ \ / \ / \ / \ / \ / \ / \ / \ / O \ / O \ / O \ / O \ / O \ / O \ o /@ o /@ o o /@ o /@ o /@ o /@ o |\ / \ /| |\ / \ / / \ / / \ /| | \ / \ / | | \ / \ / \ / \ / | | \ / \ / | | \ / O \ / O \ / O \ / | | \ / \ / | | o /@ o @\ o /@ o | | \ / \ / | | |\ / \ / \ / \ / \ /| | | \ / \ / | | | \ / \ / \ / | | | u \ / O \ / v | | u | \ / O \ / O \ / | v | o-------o @\ o-------o o---+---o @\ o @\ o---+---o \ / | \ / \ / \ / \ / | \ / | \ / \ / | \ / | du \ / O \ / dv | \ / o-------o @\ o-------o \ / \ / \ / \ / \ / \ / o o U% $T$ $E$U% o------------------>o | | | | | | | | F | | $T$F | | | | | | v v o------------------>o X% $T$ $E$X% o o / \ / \ / \ / \ / \ / O \ / \ o /@\ o / \ / \ / \ / \ / \ / \ / O \ / O \ / O \ o /@\ o o /@\ o /@\ o / \ / \ / \ \ / \ / / \ / \ / \ / \ / \ / \ / \ / \ / O \ / O \ / O \ / \ / \ o /@ o /@\ o @\ o / \ / \ / \ \ / \ / \ / \ / / \ / \ / \ / \ / \ / \ / \ / O \ / O \ / O \ / O \ / O \ / O \ o /@ o @\ o o /@ o /@ o @\ o @\ o |\ / \ /| |\ / \ / \ / \ / \ / \ /| | \ / \ / | | \ / \ / \ / \ / | | \ / \ / | | \ / O \ / O \ / O \ / | | \ / \ / | | o /@ o @ o @\ o | | \ / \ / | | |\ / / \ / \ / \ \ /| | | \ / \ / | | | \ / \ / \ / | | | x \ / O \ / y | | x | \ / O \ / O \ / | y | o-------o @ o-------o o---+---o @ o @ o---+---o \ / | \ / / \ \ / | \ / | \ / \ / | \ / | dx \ / O \ / dy | \ / o-------o @ o-------o \ / \ / \ / \ / \ / \ / o o Figure 70-a. Tangent Functor Diagram for F<u, v> = <((u)(v)), ((u, v))> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D83 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Transformations of Type B^2 -> B^2 (concl.) Figure 70-b shows another way to picture the action of the tangent functor on the logical transformation F<u, v> = <((u)(v)), ((u, v))>, roughly in the style of the "bundle of universes" type of diagram. [NB. I can't really do justice to the original Figure in ascii graphics, but this collection of pictures may serve as a construction kit, with some assembly required, to convey the general idea.] o-----------------------o o-----------------------o o-----------------------o | dU | | dU | | dU | | o--o o--o | | o--o o--o | | o--o o--o | | /////\ /////\ | | /XXXX\ /XXXX\ | | /\\\\\ /\\\\\ | | ///////o//////\ | | /XXXXXXoXXXXXX\ | | /\\\\\\o\\\\\\\ | | //////// \//////\ | | /XXXXXX/ \XXXXXX\ | | /\\\\\\/ \\\\\\\\ | | o/////// \//////o | | oXXXXXX/ \XXXXXXo | | o\\\\\\/ \\\\\\\o | | |/////o o/////| | | |XXXXXo oXXXXX| | | |\\\\\o o\\\\\| | | |/du//| |//dv/| | | |XXXXX| |XXXXX| | | |\du\\| |\\dv\| | | |/////o o/////| | | |XXXXXo oXXXXX| | | |\\\\\o o\\\\\| | | o//////\ ///////o | | oXXXXXX\ /XXXXXXo | | o\\\\\\\ /\\\\\\o | | \//////\ //////// | | \XXXXXX\ /XXXXXX/ | | \\\\\\\\ /\\\\\\/ | | \//////o/////// | | \XXXXXXoXXXXXX/ | | \\\\\\\o\\\\\\/ | | \///// \///// | | \XXXX/ \XXXX/ | | \\\\\/ \\\\\/ | | o--o o--o | | o--o o--o | | o--o o--o | | | | | | | o-----------------------o o-----------------------o o-----------------------o = du' @ (u)(v) o-----------------------o dv' @ (u)(v) = = | dU' | = = | o--o o--o | = = | /////\ /\\\\\ | = = | ///////o\\\\\\\ | = = | ////////X\\\\\\\\ | = = | o///////XXX\\\\\\\o | = = | |/////oXXXXXo\\\\\| | = = = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = = | |/////oXXXXXo\\\\\| | | o//////\XXX/\\\\\\o | | \//////\X/\\\\\\/ | | \//////o\\\\\\/ | | \///// \\\\\/ | | o--o o--o | | | o-----------------------o o-----------------------o o-----------------------o o-----------------------o | dU | | dU | | dU | | o--o o--o | | o--o o--o | | o--o o--o | | / \ /////\ | | /\\\\\ /XXXX\ | | /\\\\\ /\\\\\ | | / o//////\ | | /\\\\\\oXXXXXX\ | | /\\\\\\o\\\\\\\ | | / //\//////\ | | /\\\\\\//\XXXXXX\ | | /\\\\\\/ \\\\\\\\ | | o ////\//////o | | o\\\\\\////\XXXXXXo | | o\\\\\\/ \\\\\\\o | | | o/////o/////| | | |\\\\\o/////oXXXXX| | | |\\\\\o o\\\\\| | | | du |/////|//dv/| | | |\\\\\|/////|XXXXX| | | |\du\\| |\\dv\| | | | o/////o/////| | | |\\\\\o/////oXXXXX| | | |\\\\\o o\\\\\| | | o \//////////o | | o\\\\\\\////XXXXXXo | | o\\\\\\\ /\\\\\\o | | \ \///////// | | \\\\\\\\//XXXXXX/ | | \\\\\\\\ /\\\\\\/ | | \ o/////// | | \\\\\\\oXXXXXX/ | | \\\\\\\o\\\\\\/ | | \ / \///// | | \\\\\/ \XXXX/ | | \\\\\/ \\\\\/ | | o--o o--o | | o--o o--o | | o--o o--o | | | | | | | o-----------------------o o-----------------------o o-----------------------o = du' @ (u) v o-----------------------o dv' @ (u) v = = | dU' | = = | o--o o--o | = = | /////\ /\\\\\ | = = | ///////o\\\\\\\ | = = | ////////X\\\\\\\\ | = = | o///////XXX\\\\\\\o | = = | |/////oXXXXXo\\\\\| | = = = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = = | |/////oXXXXXo\\\\\| | | o//////\XXX/\\\\\\o | | \//////\X/\\\\\\/ | | \//////o\\\\\\/ | | \///// \\\\\/ | | o--o o--o | | | o-----------------------o o-----------------------o o-----------------------o o-----------------------o | dU | | dU | | dU | | o--o o--o | | o--o o--o | | o--o o--o | | /////\ / \ | | /XXXX\ /\\\\\ | | /\\\\\ /\\\\\ | | ///////o \ | | /XXXXXXo\\\\\\\ | | /\\\\\\o\\\\\\\ | | /////////\ \ | | /XXXXXX//\\\\\\\\ | | /\\\\\\/ \\\\\\\\ | | o//////////\ o | | oXXXXXX////\\\\\\\o | | o\\\\\\/ \\\\\\\o | | |/////o/////o | | | |XXXXXo/////o\\\\\| | | |\\\\\o o\\\\\| | | |/du//|/////| dv | | | |XXXXX|/////|\\\\\| | | |\du\\| |\\dv\| | | |/////o/////o | | | |XXXXXo/////o\\\\\| | | |\\\\\o o\\\\\| | | o//////\//// o | | oXXXXXX\////\\\\\\o | | o\\\\\\\ /\\\\\\o | | \//////\// / | | \XXXXXX\//\\\\\\/ | | \\\\\\\\ /\\\\\\/ | | \//////o / | | \XXXXXXo\\\\\\/ | | \\\\\\\o\\\\\\/ | | \///// \ / | | \XXXX/ \\\\\/ | | \\\\\/ \\\\\/ | | o--o o--o | | o--o o--o | | o--o o--o | | | | | | | o-----------------------o o-----------------------o o-----------------------o = du' @ u (v) o-----------------------o dv' @ u (v) = = | dU' | = = | o--o o--o | = = | /////\ /\\\\\ | = = | ///////o\\\\\\\ | = = | ////////X\\\\\\\\ | = = | o///////XXX\\\\\\\o | = = | |/////oXXXXXo\\\\\| | = = = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = = | |/////oXXXXXo\\\\\| | | o//////\XXX/\\\\\\o | | \//////\X/\\\\\\/ | | \//////o\\\\\\/ | | \///// \\\\\/ | | o--o o--o | | | o-----------------------o o-----------------------o o-----------------------o o-----------------------o | dU | | dU | | dU | | o--o o--o | | o--o o--o | | o--o o--o | | / \ / \ | | /\\\\\ /\\\\\ | | /\\\\\ /\\\\\ | | / o \ | | /\\\\\\o\\\\\\\ | | /\\\\\\o\\\\\\\ | | / / \ \ | | /\\\\\\/ \\\\\\\\ | | /\\\\\\/ \\\\\\\\ | | o / \ o | | o\\\\\\/ \\\\\\\o | | o\\\\\\/ \\\\\\\o | | | o o | | | |\\\\\o o\\\\\| | | |\\\\\o o\\\\\| | | | du | | dv | | | |\\\\\| |\\\\\| | | |\du\\| |\\dv\| | | | o o | | | |\\\\\o o\\\\\| | | |\\\\\o o\\\\\| | | o \ / o | | o\\\\\\\ /\\\\\\o | | o\\\\\\\ /\\\\\\o | | \ \ / / | | \\\\\\\\ /\\\\\\/ | | \\\\\\\\ /\\\\\\/ | | \ o / | | \\\\\\\o\\\\\\/ | | \\\\\\\o\\\\\\/ | | \ / \ / | | \\\\\/ \\\\\/ | | \\\\\/ \\\\\/ | | o--o o--o | | o--o o--o | | o--o o--o | | | | | | | o-----------------------o o-----------------------o o-----------------------o = du' @ u v o-----------------------o dv' @ u v = = | dU' | = = | o--o o--o | = = | /////\ /\\\\\ | = = | ///////o\\\\\\\ | = = | ////////X\\\\\\\\ | = = | o///////XXX\\\\\\\o | = = | |/////oXXXXXo\\\\\| | = = = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = = | |/////oXXXXXo\\\\\| | | o//////\XXX/\\\\\\o | | \//////\X/\\\\\\/ | | \//////o\\\\\\/ | | \///// \\\\\/ | | o--o o--o | | | o-----------------------o o-----------------------o o-----------------------o o-----------------------o | U | |\U\\\\\\\\\\\\\\\\\\\\\| |\U\\\\\\\\\\\\\\\\\\\\\| | o--o o--o | |\\\\\\o--o\\\o--o\\\\\\| |\\\\\\o--o\\\o--o\\\\\\| | /////\ /////\ | |\\\\\/////\\/////\\\\\\| |\\\\\/ \\/ \\\\\\| | ///////o//////\ | |\\\\///////o//////\\\\\| |\\\\/ o \\\\\| | /////////\//////\ | |\\\////////X\//////\\\\| |\\\/ /\\ \\\\| | o//////////\//////o | |\\o///////XXX\//////o\\| |\\o /\\\\ o\\| | |/////o/////o/////| | |\\|/////oXXXXXo/////|\\| |\\| o\\\\\o |\\| | |//u//|/////|//v//| | |\\|//u//|XXXXX|//v//|\\| |\\| u |\\\\\| v |\\| | |/////o/////o/////| | |\\|/////oXXXXXo/////|\\| |\\| o\\\\\o |\\| | o//////\//////////o | |\\o//////\XXX///////o\\| |\\o \\\\/ o\\| | \//////\///////// | |\\\\//////\X////////\\\| |\\\\ \\/ /\\\| | \//////o/////// | |\\\\\//////o///////\\\\| |\\\\\ o /\\\\| | \///// \///// | |\\\\\\/////\\/////\\\\\| |\\\\\\ /\\ /\\\\\| | o--o o--o | |\\\\\\o--o\\\o--o\\\\\\| |\\\\\\o--o\\\o--o\\\\\\| | | |\\\\\\\\\\\\\\\\\\\\\\\| |\\\\\\\\\\\\\\\\\\\\\\\| o-----------------------o o-----------------------o o-----------------------o = u' o-----------------------o v' = = | U' | = = | o--o o--o | = = | /////\ /\\\\\ | = = | ///////o\\\\\\\ | = = | ////////X\\\\\\\\ | = = | o///////XXX\\\\\\\o | = = | |/////oXXXXXo\\\\\| | = = = = = = = = = = = =|/u'//|XXXXX|\\v'\|= = = = = = = = = = = | |/////oXXXXXo\\\\\| | | o//////\XXX/\\\\\\o | | \//////\X/\\\\\\/ | | \//////o\\\\\\/ | | \///// \\\\\/ | | o--o o--o | | | o-----------------------o Figure 70-b. Tangent Functor Ferris Wheel for F<u, v> = <((u)(v)), ((u, v))> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D84 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Epilogue, Enchoiry, Exodus | It is time to explain myself . . . . let us stand up. | | Walt Whitman, 'Leaves of Grass', [Whi, 79] o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Note D85 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Appendix 1-A Operator Maps for the Disjunction "f" Table A1. Computation of "?f" ?f = f<u, v> = ((u)(v)) = u.v.f<1, 1> + u (v).f<1, 0> + (u) v.f<0, 1> + + (u)(v).f<0, 0> = u.v + u (v) + (u) v + 0 ?f = + u.v (du)(dv) + u (v) (du)(dv) + (u) v (du)(dv) + 0 + u.v (du) dv + u (v) (du) dv + (u) v (du) dv + 0 + u.v du (dv) + u (v) du (dv) + (u) v du (dv) + 0 + u.v du.dv + u (v) du.dv + (u) v du.dv + 0 Table A2. Computation of "Ef" Ef = f<u+du, v+dv> = (((u, du))((v, dv))) = u.v.f<(du), (dv)> + u (v).f<(du), dv> + (u) v.f<du, (dv)> + (u)(v).f<du, dv> = u.v (du.dv) + u (v) (du (dv)) + (u) v ((du) dv) + (u)(v)((du)(dv)) Ef = u.v (du)(dv) + u (v) (du)(dv) + (u) v (du)(dv) + 0 + u.v (du) dv + u (v) (du) dv + 0 + (u)(v) (du) dv + u.v du (dv) + 0 + (u) v du (dv) + (u)(v) du (dv) + 0 + u (v) du.dv + (u) v du.dv + (u)(v) du.dv Table A3. Computation of "Df" (1) Df = Ef + ?f = f<u+du, v+dv> + f<u, v> = (((u, du))((v, dv))) + ((u)(v)) Df = 0 + 0 + 0 + 0 + 0 + 0 + (u) v . (du) dv + (u)(v).(du) dv + 0 + u (v) . du (dv) + 0 + (u)(v).du (dv) + u.v . du.dv + 0 + 0 + (u)(v).du.dv Df = u.v . du.dv + u (v) . du (dv) + (u) v . (du) dv + (u)(v) ((du)(dv)) Table A4. Computation of "Df" (2) Df = ((u, v)) . du.dv + (v) . du (dv) + (u) . (du) dv + 0 . (du)(dv) Table A5. Computation of "df" Df = u.v . du.dv + u (v) . du (dv) + (u) v . (du) dv + (u)(v)((du)(dv)) => df = u.v . 0 + u (v) . du + (u) v . dv + (u)(v).(du, dv) Table A6. Computation of "rf" rf = Df + df Df = u.v . du.dv + u (v) . du (dv) + (u) v . (du) dv + (u)(v)((du)(dv)) df = u.v . 0 + u (v) . du + (u) v . dv + (u)(v).(du, dv) rf = u.v . du.dv + u (v) . du.dv + (u) v . du.dv + (u)(v).du.dv Table A7. Computation Summary for "f" ?f = u.v . 1 + u (v) . 1 + (u) v . 1 + (u)(v).0 Ef = u.v . (du.dv) + u (v) . (du (dv)) + (u) v . ((du) dv) + (u)(v)((du)(dv)) Df = u.v . du.dv + u (v) . du (dv) + (u) v . (du) dv + (u)(v)((du)(dv)) df = u.v . 0 + u (v) . du + (u) v . dv + (u)(v).(du, dv) rf = u.v . du.dv + u (v) . du.dv + (u) v . du.dv + (u)(v).du.dv Appendix 1-B Operator Maps for the Equality "g" Table B1. Computation of "?g" ?g = g<u, v> = ((u, v)) = u.v.g<1, 1> + u (v).g<1, 0> + (u) v.g<0, 1> + (u)(v).g<0, 0> = u.v + 0 + 0 + (u)(v) ?g = u.v (du)(dv) + 0 + 0 + (u)(v) (du)(dv) + u.v (du) dv + 0 + 0 + (u)(v) (du) dv + u.v du (dv) + 0 + 0 + (u)(v) du (dv) + u.v du.dv + 0 + 0 + (u)(v) du.dv Table B2. Computation of "Eg" Eg = g<u+du, v+dv> = (((u, du), (v, dv))) = u.v.g<(du), (dv)> + u (v).g<(du), dv> + (u) v.g<du, (dv)> + (u)(v).g<du, dv> = u.v ((du, dv)) + u (v) (du, dv) + (u) v (du, dv) + (u)(v)((du, dv)) Eg = u.v (du)(dv) + 0 + 0 + (u)(v) (du)(dv) + 0 + u (v) (du) dv + (u) v (du) dv + 0 + 0 + u (v) du (dv) + (u) v du (dv) + 0 + u.v du.dv + 0 + 0 + (u)(v) du.dv Table B3. Computation of "Dg" (1) Dg = Eg + ?g = g<u+du, v+dv> + g<u, v> = (((u, du), (v, dv))) + ((u, v)) Dg = 0 + 0 + 0 + 0 + u.v . (du) dv + u (v) . (du) dv + (u) v . (du) dv + (u)(v) . (du) dv + u.v . du (dv) + u (v) . du (dv) + (u) v . du (dv) + (u)(v) . du (dv) + 0 + 0 + 0 + 0 Dg = u.v . (du, dv) + u (v) . (du, dv) + (u) v . (du, dv) + (u)(v) . (du, dv) Table B4. Computation of "Dg" (2) Dg = 0 . du.dv + 1 . du (dv) + 1 . (du) dv + 0 . (du)(dv) Table B5. Computation of "dg" Dg = u.v . (du, dv) + u (v) . (du, dv) + (u) v . (du, dv) + (u)(v).(du, dv) => dg = u.v . (du, dv) + u (v) . (du, dv) + (u) v . (du, dv) + (u)(v).(du, dv) Table B6. Computation of "rg" rg = Dg + dg Dg = u.v . (du, dv) + u (v) . (du, dv) + (u) v . (du, dv) + (u)(v) . (du, dv) dg = u.v . (du, dv) + u (v) . (du, dv) + (u) v . (du, dv) + (u)(v) . (du, dv) rg = u.v . 0 + u (v) . 0 + (u) v . 0 + (u)(v) . 0 Table B7. Computation Summary for "g" ?g = u.v . 1 + u (v) . 0 + u (v) . 0 + (u)(v).1 Eg = u.v . ((du, dv)) + u (v) . (du, dv) + (u) v . (du, dv) + (u)(v).((du, dv)) Dg = u.v . (du, dv) + u (v) . (du, dv) + (u) v . (du, dv) + (u)(v).(du, dv) dg = u.v . (du, dv) + u (v) . (du, dv) + (u) v . (du, dv) + (u)(v).(du, dv) rg = u.v . 0 + u (v) . 0 + (u) v . 0 + (u)(v).0 Appendix 2 EF Arranged by Differential Features F EF T11F T10F T01F T00F a1 = A a2 = B S[(ai+dai)/ai]i F EF @ dA dB EF @ dA (dB) EF @ (dA) dB EF @ (dA)(dB) F0 () () () () () () F1 (A)(B) ((A,dA))((B,dB)) A B A (B) (A) B (A)(B) F2 (A) B ((A,dA)) (B,dB) A (B) A B (A)(B) (A) B F4 A (B) (A,dA) ((B,dB)) (A) B (A)(B) A B A (B) F8 A B (A,dA) (B,dB) (A)(B) (A) B A (B) A B F3 (A) ((A,dA)) A A (A) (A) F12 A (A,dA) (A) (A) A A F6 (A, B) ((A,dA), (B,dB)) (A, B) ((A, B)) ((A, B)) (A, B) F9 ((A, B)) (((A,dA), (B,dB))) ((A, B)) (A, B) (A, B) ((A, B)) F5 (B) ((B,dB)) B (B) B (B) F10 B (B,dB) (B) B (B) B F7 (A B) ((A,dA) (B,dB)) ((A)(B)) ((A) B) (A (B)) (A B) F11 (A (B)) ((A,dA) ((B,dB))) ((A) B) ((A)(B)) (A B) (A (B)) F13 ((A) B) (((A,dA)) (B,dB)) (A (B)) (A B) ((A)(B)) ((A) B) F14 ((A)(B)) (((A,dA))((B,dB))) (A B) (A (B)) ((A) B) ((A)(B)) F15 (()) (()) (()) (()) (()) (()) Total Number of Fixed Points: 4 4 4 16 DF Arranged by Differential Features DF = F + EF DF @ dA dB DF @ dA (dB) DF @ (dA) dB DF @ (dA)(dB) F0 () + () () () () () F1 (A)(B) + ((A,dA))((B,dB)) ((A, B)) (B) (A) () F2 (A) B + ((A,dA)) (B,dB) (A, B) B (A) () F4 A (B) + (A,dA) ((B,dB)) (A, B) (B) A () F8 A B + (A,dA) (B,dB) ((A, B)) B A () F3 (A) + ((A,dA)) (()) (()) () () F12 A + (A,dA) (()) (()) () () F6 (A, B) + ((A,dA), (B,dB)) () (()) (()) () F9 ((A, B)) + (((A,dA), (B,dB))) () (()) (()) () F5 (B) + ((B,dB)) (()) () (()) () F10 B + (B,dB) (()) () (()) () F7 (A B) + ((A,dA) (B,dB)) ((A, B)) B A () F11 (A (B)) + ((A,dA) ((B,dB))) (A, B) (B) A () F13 ((A) B) + (((A,dA)) (B,dB)) (A, B) B (A) () F14 ((A)(B)) + (((A,dA))((B,dB))) ((A, B)) (B) (A) () F15 (()) + (()) () () () () EF Arranged by Original Features F EF EF @ A B EF @ A (B) EF @ (A) B EF @ (A)(B) F0 () () () () () () F1 (A)(B) ((A,dA))((B,dB)) dA dB dA (dB) (dA) dB (dA)(dB) F2 (A) B ((A,dA)) (B,dB) dA (dB) dA dB (dA)(dB) (dA) dB F4 A (B) (A,dA) ((B,dB)) (dA) dB (dA)(dB) dA dB dA (dB) F8 A B (A,dA) (B,dB) (dA)(dB) (dA) dB dA (dB) dA dB F3 (A) ((A,dA)) dA dA (dA) (dA) F12 A (A,dA) (dA) (dA) dA dA F6 (A, B) ((A,dA), (B,dB)) (dA, dB) ((dA, dB)) ((dA, dB)) (dA, dB) F9 ((A, B)) (((A,dA), (B,dB))) ((dA, dB)) (dA, dB) (dA, dB) ((dA, dB)) F5 (B) ((B,dB)) dB (dB) dB (dB) F10 B (B,dB) (dB) dB (dB) dB F7 (A B) ((A,dA) (B,dB)) ((dA)(dB)) ((dA) dB) (dA (dB)) (dA dB) F11 (A (B)) ((A,dA) ((B,dB))) ((dA) dB) ((dA)(dB)) (dA dB) (dA (dB)) F13 ((A) B) (((A,dA)) (B,dB)) (dA (dB)) (dA dB) ((dA)(dB)) ((dA) dB) F14 ((A)(B)) (((A,dA))((B,dB))) (dA dB) (dA (dB)) ((dA) dB) ((dA)(dB)) F15 (()) (()) (()) (()) (()) (()) DF Arranged by Original Features DF = F + EF DF @ A B DF @ A (B) DF @ (A) B DF @ (A)(B) F0 () + () () () () () F1 (A)(B) + ((A,dA))((B,dB)) dA dB dA (dB) (dA) dB ((dA)(dB)) F2 (A) B + ((A,dA)) (B,dB) dA (dB) dA dB ((dA)(dB)) (dA) dB F4 A (B) + (A,dA) ((B,dB)) (dA) dB ((dA)(dB)) dA dB dA (dB) F8 A B + (A,dA) (B,dB) ((dA)(dB)) (dA) dB dA (dB) dA dB F3 (A) + ((A,dA)) dA dA dA dA F12 A + (A,dA) dA dA dA dA F6 (A, B) + ((A,dA), (B,dB)) (dA, dB) (dA, dB) (dA, dB) (dA, dB) F9 ((A, B)) + (((A,dA), (B,dB))) (dA, dB) (dA, dB) (dA, dB) (dA, dB) F5 (B) + ((B,dB)) dB dB dB dB F10 B + (B,dB) dB dB dB dB F7 (A B) + ((A,dA) (B,dB)) ((dA)(dB)) (dA) dB dA (dB) dA dB F11 (A (B)) + ((A,dA) ((B,dB))) (dA) dB ((dA)(dB)) dA dB dA (dB) F13 ((A) B) + (((A,dA)) (B,dB)) dA (dB) dA dB ((dA)(dB)) (dA) dB F14 ((A)(B)) + (((A,dA))((B,dB))) dA dB dA (dB) (dA) dB ((dA)(dB)) F15 (()) + (()) () () () () Differential Forms: Expanded on a Logical Basis { (dA)(dB) , dA (dB) , (dA) dB , dA dB } Alternate Notation for Terms { " , ?A , ?B , " } F DF dF F0 () 0 0 F1 (A)(B) (B) dA (dB) + (A) (dA) dB + ((A, B)) dA dB (B) ?A + (A) ?B F2 (A) B B dA (dB) + (A) (dA) dB + (A, B) dA dB B ?A + (A) ?B F4 A (B) (B) dA (dB) + A (dA) dB + (A, B) dA dB (B) ?A + A ?B F8 A B B dA (dB) + A (dA) dB + ((A, B)) dA dB B ?A + A ?B F3 (A) dA (dB) + dA dB ?A F12 A dA (dB) + dA dB ?A F6 (A, B) dA (dB) + (dA) dB ?A + ?B F9 ((A, B)) dA (dB) + (dA) dB ?A + ?B F5 (B) (dA) dB + dA dB ?B F10 B (dA) dB + dA dB ?B F7 (A B) B dA (dB) + A (dA) dB + ((A, B)) dA dB B ?A + A ?B F11 (A (B)) (B) dA (dB) + A (dA) dB + (A, B) dA dB (B) ?A + A ?B F13 ((A) B) B dA (dB) + (A) (dA) dB + (A, B) dA dB B ?A + (A) ?B F14 ((A)(B)) (B) dA (dB) + (A) (dA) dB + ((A, B)) dA dB (B) ?A + (A) ?B F15 (()) 0 0 Differential Forms: Expanded on an Algebraic Basis { 1 , dA , dB , dA dB } F DF dF F0 () 0 0 F1 (A)(B) (B) dA + (A) dB + dA dB (B) dA + (A) dB F2 (A) B B dA + (A) dB + dA dB B dA + (A) dB F4 A (B) (B) dA + A dB + dA dB (B) dA + A dB F8 A B B dA + A dB + dA dB B dA + A dB F3 (A) dA dA F12 A dA dA F6 (A, B) dA + dB dA + dB F9 ((A, B)) dA + dB dA + dB F5 (B) dB dB F10 B dB dB F7 (A B) B dA + A dB + dA dB B dA + A dB F11 (A (B)) (B) dA + A dB + dA dB (B) dA + A dB F13 ((A) B) B dA + (A) dB + dA dB B dA + (A) dB F14 ((A)(B)) (B) dA + (A) dB + dA dB (B) dA + (A) dB F15 (()) 0 0 Pointwise Differential dF = Pointwise Linear Approximation to the Difference DF dF = ?AF.dA + ?BF.dB d2F = ?AB.dA.dB dF @ A B dF @ A (B) dF @ (A) B dF @ (A)(B) F0 0 0 0 0 0 0 F1 (B) dA + (A) dB dA dB 0 dA dB dA + dB F2 B dA + (A) dB dA dB dA 0 dA + dB dB F4 (B) dA + A dB dA dB dB dA + dB 0 dA F8 B dA + A dB dA dB dA + dB dB dA 0 F3 dA 0 dA dA dA dA F12 dA 0 dA dA dA dA F6 dA + dB 0 dA + dB dA + dB dA + dB dA + dB F9 dA + dB 0 dA + dB dA + dB dA + dB dA + dB F5 dB 0 dB dB dB dB F10 dB 0 dB dB dB dB F7 B dA + A dB dA dB dA + dB dB dA 0 F11 (B) dA + A dB dA dB dB dA + dB 0 dA F13 B dA + (A) dB dA dB dA 0 dA + dB dB F14 (B) dA + (A) dB dA dB 0 dA dB dA + dB F15 0 0 0 0 0 0 Taylor Series Expansion DF = dF + d2F = ?AF.dA + ?BF.dB + ?AB.dA.dB dF @ A B dF @ A (B) dF @ (A) B dF @ (A)(B) F0 0 0 0 0 0 F1 (B) dA + (A) dB + dA dB 0 dA dB dA + dB F2 B dA + (A) dB + dA dB dA 0 dA + dB dB F4 (B) dA + A dB + dA dB dB dA + dB 0 dA F8 B dA + A dB + dA dB dA + dB dB dA 0 F3 dA dA dA dA dA F12 dA dA dA dA dA F6 dA + dB dA + dB dA + dB dA + dB dA + dB F9 dA + dB dA + dB dA + dB dA + dB dA + dB F5 dB dB dB dB dB F10 dB dB dB dB dB F7 B dA + A dB + dA dB dA + dB dB dA 0 F11 (B) dA + A dB + dA dB dB dA + dB 0 dA F13 B dA + (A) dB + dA dB dA 0 dA + dB dB F14 (B) dA + (A) dB + dA dB 0 dA dB dA + dB F15 0 0 0 0 0 Partial & Relative Differentials F ?F ?A ?F ?B dF = ?AF.dA + ?BF.dB ?A|F ?B| ?B|F ?A| F0 () 0 0 0 0 0 F1 (A)(B) (B) (A) (B) dA + (A) dB F2 (A) B B (A) B dA + (A) dB F4 A (B) (B) A (B) dA + A dB F8 A B B A B dA + A dB F3 (A) 1 0 dA F12 A 1 0 dA F6 (A, B) 1 1 dA + dB F9 ((A, B)) 1 1 dA + dB F5 (B) 0 1 dB F10 B 0 1 dB F7 (A B) B A B dA + A dB F11 (A (B)) (B) A (B) dA + A dB F13 ((A) B) B (A) B dA + (A) dB F14 ((A)(B)) (B) (A) (B) dA + (A) dB F15 (()) 0 0 0 0 0 Detail of Calculation for DF = EF + F EF @ dA dB + F @ dA dB EF @ dA (dB) + F @ dA (dB) EF @ (dA) dB + F @ (dA) dB EF @ (dA)(dB) + F @ (dA)(dB) F0 0 + 0 0 + 0 0 + 0 0 + 0 F1 A B dA dB + (A)(B) dA dB A (B) dA (dB) + (A)(B) dA (dB) (A) B (dA) dB + (A)(B) (dA) dB (A)(B) (dA)(dB) + (A)(B) (dA)(dB) F2 A (B) dA dB + (A) B dA dB A B dA (dB) + (A) B dA (dB) (A)(B) (dA) dB + (A) B (dA) dB (A) B (dA)(dB) + (A) B (dA)(dB) F4 (A) B dA dB + A (B) dA dB (A)(B) dA (dB) + A (B) dA (dB) A B (dA) dB + A (B) (dA) dB A (B) (dA)(dB) + A (B) (dA)(dB) F8 (A)(B) dA dB + A B dA dB (A) B dA (dB) + A B dA (dB) A (B) (dA) dB + A B (dA) dB A B (dA)(dB) + A B (dA)(dB) F3 A dA dB + (A) dA dB A dA (dB) + (A) dA (dB) (A) (dA) dB + (A) (dA) dB (A) (dA)(dB) + (A) (dA)(dB) F12 (A) dA dB + A dA dB (A) dA (dB) + A dA (dB) A (dA) dB + A (dA) dB A (dA)(dB) + A (dA)(dB) F6 (A, B) dA dB + (A, B) dA dB ((A, B))dA (dB) + (A, B) dA (dB) ((A, B))(dA) dB + (A, B) (dA) dB (A, B) (dA)(dB) + (A, B) (dA)(dB) F9 ((A, B))dA dB +((A, B))dA dB (A, B) dA (dB) +((A, B))dA (dB) (A, B) (dA) dB +((A, B))(dA) dB ((A, B))(dA)(dB) +((A, B))(dA)(dB) F5 B dA dB + (B) dA dB (B) dA (dB) + (B) dA (dB) B (dA) dB + (B) (dA) dB (B) (dA)(dB) + (B) (dA)(dB) F10 (B) dA dB + B dA dB B dA (dB) + B dA (dB) (B) (dA) dB + B (dA) dB B (dA)(dB) + B (dA)(dB) F7 ((A)(B))dA dB + (A B) dA dB ((A) B) dA (dB) + (A B) dA (dB) (A (B))(dA) dB + (A B) (dA) dB (A B) (dA)(dB) + (A B) (dA)(dB) F11 ((A) B) dA dB + (A (B))dA dB ((A)(B))dA (dB) + (A (B))dA (dB) (A B) (dA) dB + (A (B))(dA) dB (A (B))(dA)(dB) + (A (B))(dA)(dB) F13 (A (B))dA dB +((A) B) dA dB (A B) dA (dB) +((A) B) dA (dB) ((A)(B))(dA) dB +((A) B) (dA) dB ((A) B) (dA)(dB) +((A) B) (dA)(dB) F14 (A B) dA dB +((A)(B))dA dB (A (B))dA (dB) +((A)(B))dA (dB) ((A) B) (dA) dB +((A)(B))(dA) dB ((A)(B))(dA)(dB) +((A)(B))(dA)(dB) F15 1 + 1 1 + 1 1 + 1 1 + 1 Appendix 3 [Sophus Lie, 1880] Problem. To determine the most general function f of x and r parameters a1, a2, ... , ar satisfying an equation of the form f ( f (x, a1, ... , ar), b1, ... , br) = f (x, c1, ... , cr) in which it is assumed that the ci depend only on the a's and the b's. This problem can perhaps be more clearly formulated by using the concept of a transformation group, which we now define. Definition. A family of transformations x? = f (x, a1, ... , ar), where x? denotes the original variable, x the new one, and the ai parameters, forms a transformation group if the composition of two transformations of the family is a transformation of the family, i.e., when from the equations x? ? f (x, a1, ... , ar), x? ? f (x?, b1, ... , br), there follows x? ? f (x, c1, ... , cr), where the ci are functions of the a's and b's alone. [p. 117] If in the equations xi? ? fi (x1, ... , xn, a1, ... , ar), (i = 1, 2, ... , n), one considers x1?, ... , xn? as original variables and x1, ... , xn as new variables and a1, ... , ar as parameters, then these equations define 8r transformations. I say that such a family of transformations forms a group if the composition of two transformations of the family is again a transformation of the family, i.e., when from the equations xi? ? fi (x1, ... , xn, a1, ... , ar) = fi (a), xi? ? fi (x1?, ... , xn?, b1, ... , br), follows xi? ? fi (x1, ... , xn, c1, ... , cr), where c1, ... , cr depend only on the a's and the b's, and neither on x or the index i. In other words, we require equations fi ( f1 (a), ... , fn (a), b1, ... , br) = fi (x1, ... , xn, c1, ... , cr). [p. 171] [Sophus Lie, 1880] A transformation is said to be infinitesimal if it can be put in the form xi? = xi + Xi (x1, ... , xn).?t where ?t is an infinitesimal. Generally, we shall write such equations as ?xi = Xi (x1, ... , xn).?t . If one replaces x1, ... , xn by new variables, say y1, ... , yn, then our infinitesimal transformation assumes the form ?yi = ?t.?k ?yi/?xk X . On the other hand, if we make the same change of variables in the expression A(F) = X1.?F/?x1 + ... + Xn.?F/?xn , we get A(F) = ?F/?y1 ?k ?y1/?xk Xk + ... + ?F/?yn ?k ?yn/?xk Xk . Thus we see that the equations of the infinitesimal transformation and the expression A(F) transform in the same way. Therefore, it is analytically permissible to consider A(F) as the symbol of our infinitesimal transformation. [Lie, p. 177] Appendix 4 Various Definitions of the Tangent Vector Sources to be classified: Refs: [Sp65], [Sp79] 1st Approach. Tangent vector as differential operator. Refs: [Che46, 76], [dCa, 6-7], [Hic, 5-6, 18], [KoA, 13] 2nd Approach. Tangent vector as path classifier. Refs: [DoM] 3rd Approach. Tangent vector via tangent functor. Refs: [JGH] 4th Approach. Tangent vector as equivalence class of triples (U, ?, u), using chart maps ?, ?. (U, ?, u) ˜ (V, ?, v) if "derivative" (differential) of transition map ? ° ?-1 at ?(x) takes u to v. (? ° ?-1)? (?x) u = v Helpful hint: Read "(? ° ?-1)(x)" as: "my name for what you call x", or "the new name for what I used to call x". Refs: [La83, 489], [La85, 26], [La93, 533], [Hir, 16-17] References Works Cited [AuM] Auslander, L. & MacKenzie, R.E. Introduction to Differentiable Manifolds. 1st published: McGraw-Hill, 1963. Reprinted: Dover, New York, NY, 1977. [BiG] Bishop, R.L. & Goldberg, S.I. Tensor Analysis on Manifolds. 1st published: Macmillan, 1968. Reprinted: Dover, New York, NY, 1980. [Boo] Boole, G. An Investigation of The Laws of Thought. 1st published: Macmillan, 1854. Reprinted: Dover, New York, NY, 1958. [BoT] Bott, R. & Tu, L.W. Differential Forms in Algebraic Topology. Springer-Verlag, New York, NY, 1982. [dCa] Carmo, M.P. do. Riemannian Geometry. Originally published in Portuguese: 1st ed. 1979. 2nd ed. 1988. Translated by F. Flaherty. Birkhäuser, Boston, MA, 1992. [Che46] Chevalley, C. Theory of Lie Groups. Princeton University Press, Princeton, NJ, 1946. [Che56] Chevalley, C. Fundamental Concepts of Algebra. Academic Press, 1956. [Cho86] Chomsky, N. Knowledge of Language: Its Nature, Origin, and Use. Praeger, New York, NY, 1986. [Cho93] Chomsky, N. Language and Thought. Moyer Bell, Wakefield, RI, 1993. [DoM] Doolin, B.F. & Martin, C.F. Introduction to Differential Geometry for Engineers. Marcel Dekker, New York, NY, 1990. [Fuji] Fujiwara, H. Logic Testing and Design for Testability. MIT Press, Cambridge, MA, 1985. [Hic] Hicks, N.J. Notes on Differential Geometry. Van Nostrand, Princeton, NJ, 1965. [Hir] Hirsch, M.W. Differential Topology. Springer-Verlag, New York, NY, 1976. [How] Howard, W.A. "The Formulae-as-Types Notion of Construction". Notes circulated from 1969. Reprinted in [SeH, 479-490]. [JGH] Jones, A., Gray, A., & Hutton, R. Manifolds and Mechanics. Cambridge University Press, Cambridge, UK, 1987. [KoA] Kosinski, A.A. Differential Manifolds. Academic Press, San Diego, CA, 1993. [Koh] Kohavi, Z. Switching and Finite Automata Theory. 2nd edition. McGraw-Hill, New York, NY, 1978. [LaS] Lambek, J. & Scott, P.J. Introduction to Higher Order Categorical Logic. Cambridge University Press, Cambridge, UK, 1986. [La83] Lang, S. Real Analysis. 2nd edition. Addison-Wesley, Reading, MA, 1983. [La84] Lang, S. Algebra. 2nd edition. Addison-Wesley, Menlo Park, CA, 1984. [La85] Lang, S. Differential Manifolds. Springer-Verlag, New York, NY, 1985. [La93] Lang, S. Real and Functional Analysis. 3rd edition. Springer-Verlag, New York, NY, 1993. [Lie80] Lie, S. "Sophus Lie's 1880 Transformation Group Paper". In Lie Groups: History, Frontiers, and Applications, Volume 1. Translated by M. Ackerman. Comments by R. Hermann. Math Sci Press, Brookline, MA, 1975. Original paper 1880. [Lie84] Lie, S. "Sophus Lie's 1884 Differential Invariant Paper". In Lie Groups: History, Frontiers, and Applications, Volume 3. Translated by M. Ackerman. Comments by R. Hermann. Math Sci Press, Brookline, MA, 1976. Original paper 1884. [LoS] Loomis, L.H. & Sternberg, S. Advanced Calculus. Addison-Wesley, Reading, MA, 1968. [Mel] Melzak, Z.A. Mathematical Ideas, Modeling and Applications. Volume 2 of Companion to Concrete Mathematics. John Wiley & Sons, New York, NY, 1976. [Men] Menabrea, L.F. "Sketch of the Analytical Engine Invented by Charles Babbage" With Notes by the Translator: Ada Augusta (Byron), Countess of Lovelace. In [M&M, 225-297]. Originally published 1842. [M&M] Morrison, P. & Morrison, E. (eds.) Charles Babbage on the Principles and Development of the Calculator, and Other Seminal Writings by Charles Babbage and Others. With an Introduction by the Editors. Dover, Mineola, NY, 1961. [P1] Peirce, C.S. Collected Papers of Charles Sanders Peirce. Volumes 1-8. Edited by: C. Hartshorne, P. Weiss, & A.W. Burks. Harvard University Press, Cambridge, MA, 1931-1960. References are given as: "CP [vol.].[paragraph]". [P2] -. "Qualitative Logic". In The New Elements of Mathematics, Volume 4. Edited by: C. Eisele. Mouton, The Hague, 1976. References are given as: "NE [vol.],[page]". [Rob] Roberts, D.D. The Existential Graphs of Charles S. Peirce. Mouton, The Hague, 1973. [SeH] Seldin, J.P. & Hindley, J.R. (eds.) To H.B. Curry: Essays on Combinatory Logic, Lambda Calculus, and Formalism. Academic Press, London, UK, 1980. [SpB] Spencer-Brown, G. Laws of Form. George Allen & Unwin, London, UK, 1969. [Sp65] Spivak, M. Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus. W.A. Benjamin, New York, NY, 1965. [Sp79] Spivak, M. A Comprehensive Introduction to Differential Geometry. Volumes 1 & 2. 1st edition 1970. 2nd edition: Publish or Perish Inc., Houston, TX, 1979. [Sty] Styazhkin, N.I. History of Mathematical Logic from Leibniz to Peano. 1st published in Russian: Nauka, Moscow, 1964. MIT Press, Cambridge, MA, 1969. [Wie] Wiener, N. Cybernetics: or Control and Communication in the Animal and the Machine. 1st edition 1948. 2nd edition: MIT Press, Cambridge, MA, 1961. Works Consulted [Ami] Amit, D.J. Modeling Brain Function: The World of Attractor Neural Networks. Cambridge University Press, Cambridge, UK, 1989. [Ed87] Edelman, G.M. Neural Darwinism: The Theory of Neuronal Group Selection. Basic Books, New York, NY, 1987. [Ed88] Edelman, G.M. Topobiology: An Introduction to Molecular Embryology. Basic Books, New York, NY, 1988. [Fla] Flanders, H. Differential Forms with Applications to the Physical Sciences. Dover, Mineola, NY, 1989. First published: Academic Press, 1963. [Has] Hassoun, M.H. (ed.) Associative Neural Memories: Theory and Implementation. Oxford University Press, New York, NY, 1993. [KoB] Kosko, B. Neural Networks and Fuzzy Systems: A Dynamical Systems Approach to Machine Intelligence. Prentice-Hall, Englewood Cliffs, NJ, 1992. [MaB] Mac Lane, S. & Birkhoff, G. Algebra. 3rd edition. Chelsea, New York, NY, 1993. [Mac] Mac Lane, S. Categories for the Working Mathematician. Springer-Verlag, New York, NY, 1971. [McC] McCulloch, W.S. Embodiments of Mind. MIT Press, Cambridge, MA, 1965. [Mc1] -. "A Heterarchy of Values Determined by the Topology of Nervous Nets", Bulletin of Mathematical Biophysics, Vol. 7 (1945), pp. 89-93. Reprinted in [McC]. [MiP] Minsky, M.L. & Papert, S.A. Perceptrons: An Introduction to Computational Geometry. MIT Press, Cambridge, MA, 1969. 2nd printing 1972. Expanded edition 1988. [Rum] Rumelhart, D.E., Hinton, G.E., & McClelland, J.L. "A General Framework for Parallel Distributed Processing" = Chapter 2 in Rumelhart, McClelland, & the PDP Research Group, Parallel Distributed Processing; Explorations in the Microstructure of Cognition, Volume 1: Foundations. MIT Press, Cambridge, MA, 1986. Incidental Works [Dew] Dewey, John. How We Think. Prometheus Books, Buffalo, NY, 1991. Originally published: D.C. Heath, Lexington, MA, 1910. [Fou] Foucault, Michel. The Archaeology of Knowledge & The Discourse on Language. Translated by A.M. Sheridan-Smith & Rupert Swyer. Pantheon, New York, NY, 1972. Originally published as L'Archéologie du Savoir & L'ordre du discours, Editions Gallimard, 1969 & 1971. [Hom] Homer. The Odyssey. With an English translation by A.T. Murray. Loeb Classical Library, Harvard University Press, Cambridge, MA, 1980. First printed 1919. [Jam] James, William. Pragmatism: A New Name for Some Old Ways of Thinking. Longmans, Green, & Co., New York, NY, 1907. [Ler] Leroux, Gaston. The Phantom of the Opera. Foreword by P. Haining. Dorset Press, New York, NY, 1988. Originally published in French, 1911. [Mus] Musil, Robert. The Man Without Qualities. 3 volumes. Translated with a Foreword by Eithne Wilkins and Ernst Kaiser. Pan Books, London, UK, 1979. English edition first published by Secker & Warburg, 1954. Originally published in German: Der Mann ohne Eigenschaften, 1930 & 1932. [PlaR] Plato. The Republic. With an English translation by Paul Shorey. Loeb Classical Library, Harvard University Press, Cambridge, MA, 1980. First printed 1930 & 1935. [PlaS] Plato. The Sophist. Loeb Classical Library, William Heinemann, London, 1921, 1987. [Qui] Quine, W.V. Mathematical Logic. 1st edition, 1940. Revised edition, 1951. Harvard University Press, Cambridge, MA, 1981. [SaD] Santillana, Giorgio de, & Dechend, Hertha von. Hamlet's Mill: An Essay on Myth and the Frame of Time. David R. Godine, Publisher, Boston, MA, 1977. 1st published 1969. [Sha] Shakespeare, Wm. William Shakespeare: The Complete Works. Compact Edition. General editors: S. Wells & G. Taylor. Oxford University Press, Oxford, UK, 1988. [Sh1] -. A Midsummer Night's Dream. Washington Square Press, New York, NY, 1958. [Sh2] -. The Tragedy of Hamlet, Prince of Denmark. In [Sha], pp. 654-690. [Sh3] -. Measure for Measure. Washington Square Press, New York, NY, 1965. [Web] Webster's Ninth New Collegiate Dictionary. Merriam-Webster, Springfield, MA, 1983. [Whi] Whitman, Walt. Leaves of Grass. Vintage Books / The Library of America, New York, NY, 1992. Originally published in numerous editions, 1855-1892. [Wil] Wilhelm, R. (trans.) The I Ching, or Book of Changes. Translated by R. Wilhelm & C.F. Baynes. Foreword by C.G. Jung. Preface by H. Wilhelm. 3rd edition: Bollingen Series XIX, Princeton University Press, Princeton, NJ, 1967. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Work Area o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o problem about writing e = e:e + f:f + g:g + h:h no recursion intended need for a work-around ways way explaining it away action on signs not objects math def of rep o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Old Version o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Table 4. A Polymorphous Function _____________________________ | | | | | T | |_____________________|_______| | | | | A B C | t | | A B ( C ) | t | | A ( B ) C | t | | A ( B ) ( C ) | f | | ( A ) B C | t | | ( A ) B ( C ) | f | | ( A ) ( B ) C | f | | ( A ) ( B ) ( C ) | f | |_____________________|_______| With these preliminaries we are ready to see how the form of expression and means of evaluation used in our version of ExG (Existential Graphs) may be related to the usual pictures. Looking at the initial columns of the truth table, and having seen that the relation between the heading and the rows is one of interpretation, we can let the rows suffice to interpret themselves by replacing their 1 and 0 values with their "A" and "not A" positions regarding each proposition A in the column heads. In our current example, this gives the arrangement in Table 4. Here we have used the one-slot operator to express "not A" by "( A )". Also, because the parser that we use in Study will later require it, we have adopted the habit of terminating each logical term (even those of one symbol) with a blank space. | Note for later use: | | The parser for log files will accept <space> and | also <end of line> or <carriage return> characters to | terminate a word, but not <end of file> or <control-z> | characters. Extra spaces and lines, not within a word, | may be added freely. Figure 3. A Polymorphous Function T : \ ____ A ___ B ___ C * | | | | | |__ ( C ) * | | | |__ ( B ) _ C * | | | |__ ( C ) - | |__ ( A ) _ B ___ C * | | | |__ ( C ) - | |__ ( B ) _ C - | |__ ( C ) - In our next transformation of the Truth table we take the rows in their present order as though they were the word sequences in some language, treating strings of the form "A" and "( A )" as distinct words, and then we unify common initial segments of these Strands to form a tree. As the leaves of this tree we take the truth functional values of the proposition, here signifying "true" by "*" and "false" by "-". This gives us the tree form shown in Figure 3. Figure 4. A Polymorphous Function _____________________________ | | | A | | B * | | (B ) | | C * | | (C ) - | | (A ) | | B | | C * | | (C ) - | | (B ) - | |_____________________________| Finally we compress this tree into an outline form, as shown in Figure 4. This is the actual output of the Model function in the Study section when put to work on a log file containing the proposition: (( A B )( B C )( C A )), that is, { A and B } or { B and C } or { C and A }. The process of making up log files will be taken up shortly, but first we need to discuss some unexplained aspects of the outline form just given. You have probably noticed that the outline produced for our current example did not cover all of the branches of the preceding tree in equal detail or list all of its possible paths. Namely, it summarizes only the following: A B * A (B ) C * A (B ) (C ) - (A ) B C * (A ) B (C ) - (A ) (B ) - Thus, we can see that the tree actualized by Model has only six leaves in comparison to the eight that were possible. This is because the Model function has taken each path only as far as necessary to be certain of its value under the truth function given. In this case, the first and last paths did not have to be taken through their final branch because their values were already detectable. This is exactly the kind of advantage in efficiency that we want to make the maximum possible use of, since the work of exploring 2^N rows of a table or paths of a tree becomes quickly prohibitive. But there are limits to how well we can do. Some expressions are just so complex, and there is a minimum complexity associated with all equivalent ways of expressing them. For many of the natural examples however, the universes of discourse we actually find ourselves needing to model, things are usually not so bad. In these situations we can treat Propositional Calculus as a very simple type of declarative programming language, not too rich in what it can express but useful at any rate for certain jobs, with the Model function acting as the evaluator or interpreter of the language. Seen in this light it is not so surprising that there can be arbitrarily inefficient programs for the same task, the question is whether we can find better ones. For many of the problems that arise spontaneously in prctice, and this can only be an empirically defined class for now, it often happens that more efficient descriptions can be found literally by adding more variables, so long as these are used to supply more constraining information on the space of models. There is one other way that we can improve the procedure for evaluating interpretations, that is, for finding the value of a proposition at the end of a table row, a tree path, or a sequence of indented topic headings in an outline. In the original truth table, each row must list the variables in the same order, but now that we have given each interpretation its own head we can let them ride off in all directions at once. In other words, once two roads have diverged they need not visit the remaining places in the same order, not when some advantage can be seen to doing otherwise. These remarks will become clearer when we see some larger examples where the Model function has room to exploit this strategy. Finally, before we leave our polymorphous example, there is one last aspect of the Model function that we need to discuss, the subject of Normal Forms. We have seen the concept of a logical model appear in several guises: the true interpretations, the rows of a truth table or the paths of a tree that lead to a true value or that satisfy a proposition, the cells of an indicated region in a Venn diagram. Notice that we can use the set of models of a proposition to obtain a logically equivalent proposition whose meaning across cases is manifestly clear, being expressed by independent pieces. We do this by transforming each model into the conjunction of its features and then forming the disjunction of the whole set of such conjuncts. The result is called a Disjunctive Normal Form (DNF). In our present example we get: o-----------------------o o--------------------------------o | | | | | (( A B | which | either { A and B } | | )( A (B ) C | means: | or { A and -B and C } | | )( (A ) B C | | or { -A and B and C } | | )) | | <end> | | | | | o-----------------------o o--------------------------------o Here we illustrate how the log file parser lets us arrange the disjuncts on separate lines, and use extra spaces for ease of reading. The DNF that Model gave us here is probably not too impressive, since the initial expression was already in such a form, and a shorter one at that -- still, it does the job. There are a number of other functions in the Study section, all of which start with the normal form produced by Model and derive various abstracts of it which, depending on the proposition, may be complete enough and even more clear for most purposes. Rather than think up new qualifiers (like pseudo-, quasi-, semi-) for these new forms, we will simply refer to them all as "normal forms" for now. With this we arrive at our ultimate description of the type of logical modeling process implemented here. In sum, Study is a set of functions for computing normal forms of propositions. Their essential character is that of clarifying expression while preserving meaning. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | References | | Angluin, Dana, |"Learning with Hints", in: |'Proceedings of the 1988 Workshop on Computational Learning Theory', | edited by D. Haussler & L. Pitt, Morgan Kaufmann, San Mateo, CA, 1989. | | Peirce, C.S., |'Collected Papers of Charles Sanders Peirce', | edited by C. Hartshorne, P. Weiss, & A.W. Burks, 8 Volumes, | Harvard University Press, Cambridge, MA, 1931-1960. | Cited in the form: CP Volume.Paragraph. | | Spencer Brown, George, (1969), |'Laws of Form', George Allen & Unwin, London, UK, 1969. | | Edelman, Gerald M., (1988), |'Topobiology: An Introduction to Molecular Embryology', | Basic Books, New York, NY, 1988. | | McClelland, J.L. & Rumelhart, D.E., (1988), |'Explorations in Parallel Distributed Processing: | A Handbook of Models, Programs, and Exercises', | MIT Press, Cambridge, MA, 1988. | | Maier, D. & Warren, D.S., |'Computing with Logic: Logic Programming with Prolog', | Benjamin/Cummings, Menlo Park, CA, 1988. | | Denning, P.J., Dennis, J.B., & Qualitz, J.E., |'Machines, Languages, & Computation', | Prentice-Hall, Englewood Cliffs, NJ, 1978. | | Lloyd, J.W., |'Foundations of Logic Programming', | Springer-Verlag, Berlin, Germany, 1984. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o One way to do this is as follows. Look at the logical features that are mentioned in a proposition or in a set of propositions, and take down their names as one's initial or minimal "alphabet" of logical variable names, for example: | For the single proposition, | | q = (( u v )( u w )( v w )), | | adopt the initial alphabet: | | #X# = {"u", "v", "w"}. Given an "alphabet" #X# = {"x<1>", ..., "x<k>"}, whose terms may be interpreted as names of logical features a<j>, for j = 1 to k, we can use this as a basis to construct a concrete logical space, notated as X = <#X#>, that can be defined in the following steps: 1. Define the "i^th coordinate dimension" as X<i> = {"(x<i>)", "x<i>"}. For the moment, this is just a set that consists of two expressions. As interpreted, these expressions are particular ways of referring to values in B, where "(x<i>)" denotes the negation of the value of x<i>, and where "x<i>" denotes the value of x<i>, respectively. 2. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Discussion Notes A o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Discusssion Note A1 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o BU = Benjamin Udell BU: Your exposition of differential logic is over my head, YET -- Apologies to all for posting so many notes at once, but I've found that it's best to break this stuff up into easy pieces, and I wanted to get to the part about the pragmatic maxim before everyone lapsed into a coma. Too late, most likely. I just thought that it was about time that I supply a concrete example in support of all those wild claims I've been making about how crucial Peirce's mathematical way of looking at logic is to the future of both subjects. From my perspective, his logic is not some museum curiosity, but a living force and a working tool, a resource whose full potential is yet to be fully explored. By way of illustrating the power of this approach, I will exposit here the subject of differential logic along lines that a slight extension of Peirce's Alpha Graphs makes possible. The basic idea of differential logic was hinted at by Leibniz, exists in explicit form as far back as the Boole-DeMorgan correspondence, it was familiar to Babbage, and is well-known to circuit engineers today, but its full development has been hobbled by the recalcitrant calculus with which today's logic teachers still shackle today's logic students. BU: I'm wondering whether you could do me (or maybe a few of us) the favor of temporarily morphing into E.T. Bell & explaining to a mathematically ill educated person like me, what differential logic involves. (E.g., does this have something to do with 1st- vs. 2nd-order logic?) I also mean analogously as in the following examples: Oh gee, could I play John Taine instead? Bell was a bit notorious for tailoring the facts as befit the better story. We are building the differential extension of "Zeroth Order Logic" (ZOL), that is to say, starting with propositional calculus or sentential logic. BU: Ex.: Measure theory is used for probability theory. The basic thing is to find the relative sizes of different portions of the area under the curve (the total area is usually set at unity). (If I've got that right!) This is finding the definite integrals representing the portions. (Actually I've probably got this wrong.) This is square measure theory in a venn diagram world. You may find it useful to stroll through this gallery: http://suo.ieee.org/ontology/msg03585.html BU: Ex.: In optimization sometimes one looks for the minimum or maximum of a curve. This amounts to finding the point(s) of the curve where the slope is zero. Sometimes one wants to find the intersections of various curves; in any case sometimes one seeks to find points on curves, points which have certain specified properties in terms of the curve, such as being a minimum, a maximum, a point of intersection, etc. The slope is a derivative, df/dx, which is a number in the relevant field, being the coefficient that sits next to the differential factor dx in the appropriate differential expansion. It turns out to be a bit more useful to preserve the whole differential term. Since our field is B = GF(2), the derivative is either 0 or 1, so the term dx is either there or not. BU: I do see you mentioning finding of differentials, but I don't know whether that's the basic point. Also, I'm a little confused in my ignorance, since I thought that if you're talking about discrete objects (statements), you'd be talking about differences rather than differentials. In any case I'm not sure how to think about a differential or a difference between two statements. We are starting with the logical analogue of "finite difference calculus", and will work up to the logical analogue of true differentials bit by bit. The definition that you want to keep in mind is the concept of a differential as a locally linear approximation to a function. This is a notion that can very often make sense even when all of the familiar formulas for it fail to carry over by means of the usual brands of automatic analogues. Think of a proposition, a shaded region in a venn diagram, as if the shaded region were a mesa of height 1, and view that as a potential function or a probability density on the universe of discourse. Then think about gradients. To be continuous --- If not exactly Uniformly ... o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Discusssion Note A2 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o BU = Benjamin Udell JA = Jon Awbrey BU: I found this at Semeion, Research Center of Sciences of Communication: http://www.semeion.it/GLOSSTH1.htm | Differential Logic: is a different logic to build up | complex systems. Its inspiration is biology. According | to the differential logic, a unit develops dividing itself | into more units and, in doing so, radically changes the state | of its information. This logic is not tautological, because | during the process the systems increases its quantity of | organization. | Differential System: is a system whose development happens in the same | way as biological systems; that is, through differentiation of its units. BU: Is this the same differential logic that you're talking about? I think that they are speaking of "differentiation" in the sense of embryology or developmental biology. That happens to be a big interest of mine in a remotely related way -- the data structures, one of whose alternate nicknames is "conifers", that I use in my "learning and reasoning" program, were partly influenced by the way that so-called "growth cones" ramify throughout the nervous system in the development of neural tissue during neurogenesis and epigenetic learning. Other than that, there's no terribly close conscious connection with what I'm doing with diff log at the moment. JA: We are building the differential extension of "Zeroth Order Logic" (ZOL), that is to say, starting with propositional calculus or sentential logic. BU: Ex.: Measure theory is used for probability theory. The basic thing is to find the relative sizes of different portions of the area under the curve (the total area is usually set at unity). (If I've got that right!) This is finding the definite integrals representing the portions. (Actually I've probably got this wrong.) JA: This is square measure theory in a venn diagram world. You may find it useful to stroll through this gallery: JA: http://suo.ieee.org/ontology/msg03585.html BU: If it's square measure theory, ultimately the interest will be in some kind of logical analog of mathematical integration? I just mean that propositions are (modeled as, regarded as) step-functions, functions having the form f : X -> B, where X is the universe of discourse and B = {0, 1}. If B is regarded as a "field", a space with some analogue of the usual four functions (add, subtract, multiply, divide), then it is called the "galois field of order 2" and notated as GF(2). In set theory these are called "characteristic functions" and in statistics they are known as "indicator functions" because they characterize or indicate the subset of X where f evaluates to 1. This subset is the inverse image of 1 under f, horribly notated in Asciiland as f^(-1)(1) c X, and various other folks call it the "antecedent", the "fiber", or the "pre-image" of 1 under f. I tend to use the "fiber" language, and also make use of the "fiber bars" [|...|] that allow of the more succinct form [| f |] = f^(-1)(1) = {x in X : f(x) = 1}. B ^ 1 + ****** *********** | * * * * | * * * * 0 o*****----******---------***********> X BU: Ex.: In optimization sometimes one looks for the minimum or maximum of a curve. This amounts to finding the point(s) of the curve where the slope is zero. Sometimes one wants to find the intersections of various curves; in any case sometimes one seeks to find points on curves, points which have certain specified properties in terms of the curve, such as being a minimum, a maximum, a point of intersection, etc. In mathematics one tends to take spaces and the functions on spaces in tandem, considering ordered pairs like (X, X -> K), where X is the space of interest, K is a space with a special relation to X, typically its "field of scalars", and (X -> K) is the set of all pertinent functions from X to K. In differential logic, we try to exploit what analogies we can find between real settings like (X, X -> R) and boolean settings like (Y, Y -> B), where R is the set of real numbers and B = {0, 1}. At the entry level of generality, standard tricks of the trade permit us to "coordinate" X as a k-dimensional real space R^k and Y as a k-dimensional boolean space B^k, and so we begin by cranking the analogy mill forth and back between (R^k, R^k -> R) and (B^k, B^k -> B). Starting to nod off ... will have to get to the rest tomorrow. JA: The slope is a derivative, df/dx, which is a number in the relevant field, being the coefficient that sits next to the differential factor dx in the appropriate differential expansion. It turns out to be a bit more useful to preserve the whole differential term. Since our field is B = GF(2), the derivative is either 0 or 1, so the term dx is either there or not. BU: Huh? BU: I do see you mentioning finding of differentials, but I don't know whether that's the basic point. Also, I'm a little confused in my ignorance, since I thought that if you're talking about discrete objects (statements), you'd be talking about differences rather than differentials. In any case I'm not sure how to think about a differential or a difference between two statements. JA: We are starting with the logical analogue of "finite difference calculus", and will work up to the logical analogue of true differentials bit by bit. JA: The definition that you want to keep in mind is the concept of a differential as a locally linear approximation to a function. This is a notion that can very often makes sense even when all of the familiar formulas for it fail to carry over by means of the usual brands of automatic analogues. JA: Think of a proposition, a shaded region in a venn diagram, as if the shaded region were a mesa of height 1, and view that as a potential function or a probability density on the universe of discourse. Then think about gradients. BU: potential function? gradients? o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Discusssion Note A3 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o IS = Inna Semetsky JA = Jon Awbrey IS: You mentioned circuit engineers in one of your posts. Computer technology is based on designing circuits aiming at information processing. With this in mind, how then Peirce's philosophy differ from the so called computational brand of contemporary cognitive science who equate "mind" with the information processing device, and posit that there is nothing else to it. That discussion was rendered a hopeless muddle by the fact that cognitive science folks never read anything beyond a ten-year window on their own literature, if that much, and so they fell into using the term "functionalism" in a way that was almost exactly the opposite of the way that it had always been used before. At any rate, the interesting part of the Whole Idea goes back to Aristotle's dictum that "soul is form", In that form it might be something worth discussing. IS: Indeed difference may be considered as an "error" between input and output, and manipulated upon by further differentiations to feed into "the process" again and again. I was very impressed with your posts on differential logic (I admit that I just skimmed them) but couldn't help thinking that all this "and", "or", "if ... then", and other functions of Boolean algebra indeed can be, and are being, constructed electronically. Yet I would hate to think that what cognitivists are doing -- even unknowingly -- is employing Peirce's semiotics. They use Boolean logic alright. Is it all that is there in Peirce? There is a differential aspect to inquiry. Inquiry begins with uncertainty, a condition of high cognitive entropy, if you will. Differences generalize to distributions. The more uniform the distribution the higher the entropy. Uncertainties are commonly associated with several categories of difference: 1. A difference between expectation and observation is called a "surprise". 2. A difference between intention and observation is called a "problem". 3. A difference between expectation and intention is called a (I forget). The cybernetic notion of an error-controlled regulator is a special case of this. These are some of the main reasons that I thought a differential logic was needed. IS: While on the subject: I mentioned not once that part of my research is a peculiar connection between Deleuze philosophy and american pragmatism, not the least of which is the notion of difference. Deleuze has been designated as "difference engineer" and his major opus is called "Difference and Repetition". Five or six years ago, while taking a bit of a break from my normal routine, I'd started on a collection of readings along these very lines, mostly just picking them out by free association: Deleuze, 'Difference and Repetition', 'The Fold'; Derrida, 'Writing and Difference'; Lyotard, 'The Differend'; Giroux, 'Border Crossings', and so on. But I have no really clear sense of what it was all about any more. A lot of this writing always strikes me as very insightful and intuitive, while I am reading it, and then the next one says something radically different, that also strikes me as very insightful and intuitive, so after a while I tend to become just a little indifferent. But I see that I have long passages marked in the margins of the 'The Fold', so perhaps the Leibniz link is something that I will have recourse to again. Of course, 'Timaeus' and Kierkegaard 'On Repetition' are eternal favorites. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Omitted Material A o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Continuing to draw on the reduced example of group representations, I would like to draw out a few of the finer points and problems of regarding the maxim of pragmatism as a principle of representation. Let us revisit the example of an abstract group that we had befour: 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 I presented the regular post-representation of the four-group V_4 in the following form: 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 This presents the group in one big bunch, and there are occasions when one regards it this way, but that is not the typical form of presentation that we'd encounter. More likely, the story would go a little bit like this: I cannot remember any of my math teachers ever invoking the pragmatic maxim by name, but it would be a very regular occurrence for such mentors and tutors to set up the subject in this wise: Suppose you forget what a given abstract group element means, that is, in effect, 'what it is'. Then a sure way to jog your sense of 'what it is' is to build a regular representation from the formal materials that are necessarily left lying about on that abstraction site. Working through the construction for each one of the four group elements, we arrive at the following exegeses of their senses, giving their regular post-representations: 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 So if somebody asks you, say, "What is g?", you can say, "I don't know for certain but in practice its effects go a bit like this: Converting e to g, f to h, g to e, h to f". I will have to check this out later on, but my impression is that Peirce tended to lean toward the other brand of regular, the "second", the "left", or the "ante-representation" of the groups that he treated in his earliest manuscripts and papers. I believe that this was because he thought of the actions on the pattern of dyadic relative terms like the "aftermath of". Working through this alternative for each one of the four group elements, we arrive at the following exegeses of their senses, giving their regular ante-representations: e = e:e + f:f + g:g + h:h f = f:e + e:f + h:g + g:h g = g:e + h:f + e:g + f:h h = h:e + g:f + f:g + e:h Your paraphrastic interpretation of what this all means would come out precisely the same as before. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Erratum Oops! I think that I have just confounded two entirely different issues: 1. The substantial difference between right and left regular representations. 2. The inessential difference between two conventions of presenting matrices. I will sort this out and correct it later, as need be. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o I have been planning for quite some time now to make my return to Peirce's skyshaking "Description of a Notation for the Logic of Relatives" (1870), and I can see that it's just about time to get down tuit, so let this current bit of rambling inquiry function as the preamble to that. All we need at the present, though, is a modus vivendi/operandi for telling what is substantial from what is inessential in the brook between symbolic conceits and dramatic actions that we find afforded by means of the pragmatic maxim. Back to our "subinstance", the example in support of our first example. I will now reconstruct it in a way that may prove to be less confusing. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | Bein' on the twenty-third of June, | As I sat weaving all at my loom, | Bein' on the twenty-third of June, | As I sat weaving all at my loom, | I heard a thrush, singing on yon bush, | And the song she sang was The Jug of Punch. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | In the beginning was the three-pointed star, | One smile of light across the empty face; | One bough of bone across the rooting air, | The substance forked that marrowed the first sun; | And, burning ciphers on the round of space, | Heaven and hell mixed as they spun. | | Dylan Thomas, "In The Beginning", Verse 1 I'm afrayed that this thread is just bound to keep encountering its manifold of tensuous distractions, but I'd like to try and return now to the topic of inquiry, espectrally viewed in differential aspect. Here's one picture of how it begins, one angle on the point of departure: o-----------------------------------------------------------o | | | | | o-------------o | | / \ | | / \ | | / \ | | / \ | | / \ | | o o | | | | | | | | | | | Observation | | | | | | | | | | | o--o----------o o----------o--o | | / \ \ / / \ | | / \ d_I ^ o ^ d_E / \ | | / \ \/ \/ / \ | | / \ /\ /\ / \ | | / \ / @ \ / \ | | o o--o---|---o--o o | | | | | | | | | | | v | | | | | Expectation | d_O | Intention | | | | | | | | | | | | | | | o o o o | | \ \ / / | | \ \ / / | | \ \ / / | | \ o / | | \ / \ / | | o-------------o o-------------o | | | | | o-----------------------------------------------------------o From what we must assume was a state of "Unconscious Nirvana" (UN), since we do not acutely become conscious until after we are exiled from that garden of our blissful innocence, where our Expectations, our Intentions, our Observations all subsist in a state of perfect harmony, one with every barely perceived other, something intrudes on that scene of paradise to knock us out of that blessed isle and to trouble our countenance forever after at the retrospect thereof. The least disturbance, it being provident and prudent both to take that first up, will arise in just one of three ways, in accord with the mode of discord that importunes on our equanimity, whether it is Expectation, Intention, Observation that incipiently incites the riot, departing as it will from congruence with the other two modes of being. In short, we cross just one of the three lines that border on the center, or perhaps it is better to say that the objective situation transits one of the chordal bounds of harmony, for the moment marked as d_E, d_I, d_O to note the fact one's Expectation, Intention, Observation, respectively, is the mode that we duly indite as the one that's sounding the sour note. A difference between Expectation and Observation is experienced as a "Surprise", a phenomenon that cries out for an Explanation. A discrepancy between Intention and Observation is experienced as a "Problem", of the species that calls for a Plan of Action. I can remember that I once thought up what I thought up an apt name for a gap between Expectation and Intention, but I cannot recall what it was, nor yet find the notes where I recorded it. At any rate, the modes of experiencing a surprising phenomenon or a problematic situation, as described just now, are already complex modalities, and will need to be analyzed further if we want to relate them to the minimal changes d_E, d_I, d_O. Let me think about that for a little while and see what transpires. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | In the beginning was the pale signature, | Three-syllabled and starry as the smile; | And after came the imprints on the water, | Stamp of the minted face upon the moon; | The blood that touched the crosstree and the grail | Touched the first cloud and left a sign. | | Dylan Thomas, "In The Beginning", Verse 2 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | In the beginning was the mounting fire | That set alight the weathers from a spark, | A three-eyed, red-eyed spark, blunt as a flower; | Life rose and spouted from the rolling seas, | Burst in the roots, pumped from the earth and rock | The secret oils that drive the grass. | | Dylan Thomas, "In The Beginning", Verse 3 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Omitted Material B o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | Cast of Characters | | @A@ = attic A = greek alpha, | #A# = bold A, | $A$ = curly A = script A, | !A! = singly underscored A, | %A% = doubly underscored A, | x^j = x superscript j, | x_j = x subscript j. | | The part of @a@, the lower case greek, | especially in the mathematical scenes, | is often played by the underscore !a!. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Speaking of initial conditions -- Though, to speak in truth, if come the day, when do such as we, or all the likes of us, ever truly speak of our initial conditions, sensitive toward them how ever may we grow? So let me then speak of initial conditions in just the way I wit then I ultimately do, in medias res, in thick midsts of the plot. For my part there is much that begins here: [Ashby Quote] I will now make yet another attempt to re-introduce this subject to which I have alluded before under some of the following names: "Analytic Differential Ontology" (ADO), when we imagine it to be a question of being itself, or the "Differential Extension" (DE) of logic, for now, just the propositional or the sentential type of logic, specifically, for my part, as expressed in what I call a "Reflective Extension" (RE) of Peirce's Logical Graphs, in sum, named "DiffLog" and "RefLog", respectively. I will apologize at this point for all of the "rude mnemonic mechanics", but I think that you can probably guess by now that this is utterly the only way that I have found to keep track of all this stuff in my head. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o To make the present discussion more self-contained, I will need to insert my discussion of an example from the psych research literature of the times, what was known in the cognitive science, immune system models, and neural net circles as a "polymorphous set". In some parts of this old discussion, I find myself occasionally guilty of engaging in the rather sloppy practice of using the same symbol for any of the correlated notions of a concept, a set, or a truth function. Please be forgiving of this less sad and less wise person who did this. I have tried to go back through the text, marking the concept symbols in bold letters, as #Q#, the proposition names or truth-function names in plain lower case, as q, and the set names in plain capitals, as Q, but I may have missed a few, and I sometimes omit the extra markings in figures and tables. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Notation In the process of editing this uncolorized version, I have tried to fix several problems with notation that are begining to become a bother in this plain courier context, like the excessive "busy-ness" of the letter "B" being used not only for the boolean domain B = {0, 1}, but also for a logical variable. I am also switching to the more usual mathematical convention of designating the more global space of one's immediate interest with the letter "X", thus leaving the letters "U", "V", "W", and so on, free to designate the more local and transitory patches of spaces that will come to occupy one's attention in the moment to moment concern with the situation that is given as the site of one's manifold senses. To save capital letters for sets and spaces, in so far as it may be convenient, I will default to the use of lower case letters for variables or indices whenever possible, especially in abstract examples. Just as an experiment, as I am not too sure of how it will work out if pursued in a persistent manner, whenever a sign like "u" serves as a logical index, by which I mean the index of a logical proposition, in which case you may notice that it is very usual to treat "u" as being subject to the extra reading by which it can also denote a function, that is to say, a proposition of the form u : X -> B = {0, 1}, then I will invest the associated capital with the meaning of the antecedent pre-image of 1 under the function, what some call the "fiber of truth" in u, in summary, accordingly, writing "U = (u^(-1))(1)". Finally, notice that we have two ways of referring to particular points of a logical space X, what we may on divers occasions elect to treat as cells in a venn diagram, as boolean vectors in a coordinate space B^k, or as rows of values in a truth tableau, namely, we can indicate such a particular point as the vector space sum, x = <x1, x2, ..., xk>, or as the conjunctive product x = x1 x2 ... xk, and so I will quite freely switch between these two forms. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o The next thing that we typically do is consider the effects of various operators on our target proposition -- and, by the way, not too coincidentally, I will need to shift our paradigm from the psychologist's to what is more akin to the mathematician's way of using the terms "source" and "target" -- and so I shall begin to speak about the "operand" or the "source" proposition, instead of the "target" proposition as I have been up till now. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Subj: Differential Logic Date: Fri, 06 Jun 2003 09:24:06 -0400 From: Jon Awbrey <jawbrey@oakland.edu> To: SUO <standard-upper-ontology@ieee.org> CC: Inquiry <inquiry@stderr.org>, SemioCom <gdsemiocom@univ-perp.fr> One way to think about mappings between different ontologies, and also about ontologies that develop through time -- the two problems are intimately related -- is in terms of transformations from universe of discourse to universe of discourse, the sort of thing that one is naturally tempted to call a "transformation of discourse". Re-starting from the ground up, as experience constantly teaches that we must, we may contemplate simple propositions and unanalyzed predications of the sort that one finds pictured in euler-venn diagrams and that one computes in terms of bits and boolean functions. So, working at the basement level, a mapping between two different universes of discourse, which may of course be only two different ways of describing the same universe of discourse, could be written as F : U -> V, and a transformation that describes the changes that occur in a single universe of discourse, which may of course come in the corresponding varieties of "alias" and "alibi" flavors, could be written as F : U -> U. Transformations like these can be very complex things to think about -- for instance, one may be thinking of a neuroid system of formal neurons that carry one bit each, and so one's universe is a state space U that is isomorphic to B^n, where B = {0, 1} and n is roughly 10^10, give or take -- so we usually end up having to approach such creatures, the transformations F : U -> U and F : X -> Y, in a series of increasing orders of approximation. That is what differential calculus is all about. A "derivative" or a "differential" of a transformation F is a "locally linear approximation" to F. In many ways, one can think of differentiation as an operation that takes the global description of a transformation and distributes the information into locally relevant forms. These days, differential calculus and differential geometry are carried out in terms of a thing called the "tangent functor", which is the category theoretic expression of what we do when we take derivatives. A "functor" W is a "mapping of maps" or a "transformation of transformations", so W takes a map F : X -> Y into another map WF : WX -> WY. Roughly speaking then, the particular sort of functor that we will soon know and love as the "tangent functor" T is one that takes the map F : X -> Y and gives back the locally relevant version TF : TX -> TY. Cranking the analogy for logic produces the subject of "differential logic", which has been my pursuit for a decade or two. I am almost done serializing one of my more coherent, but also more detailed, papers on the subject, and I have appended the outline of links so far. [DLOG D. Notes 01-81]

## Differential Logic 2003–2004 • Document History

### Differential Logic • SUO List • History

- http://web.archive.org/web/20071009001047/http://suo.ieee.org/email/msg09543.html
- http://web.archive.org/web/20070705085032/http://suo.ieee.org/email/msg09622.html
- http://web.archive.org/web/20070705085032/http://suo.ieee.org/email/msg10327.html

### Differential Logic • Series A • History

#### DLOG A • Ontology List

- http://web.archive.org/web/20140406040004/http://suo.ieee.org/ontology/msg04040.html
- http://web.archive.org/web/20110612001949/http://suo.ieee.org/ontology/msg04041.html
- http://web.archive.org/web/20110612010502/http://suo.ieee.org/ontology/msg04045.html
- http://web.archive.org/web/20110612005212/http://suo.ieee.org/ontology/msg04046.html
- http://web.archive.org/web/20110612001954/http://suo.ieee.org/ontology/msg04047.html
- http://web.archive.org/web/20110612010620/http://suo.ieee.org/ontology/msg04048.html
- http://web.archive.org/web/20110612010550/http://suo.ieee.org/ontology/msg04052.html
- http://web.archive.org/web/20110612010724/http://suo.ieee.org/ontology/msg04054.html
- http://web.archive.org/web/20110612000847/http://suo.ieee.org/ontology/msg04055.html
- http://web.archive.org/web/20110612001959/http://suo.ieee.org/ontology/msg04067.html
- http://web.archive.org/web/20110612010507/http://suo.ieee.org/ontology/msg04068.html
- http://web.archive.org/web/20110612002014/http://suo.ieee.org/ontology/msg04069.html
- http://web.archive.org/web/20110612010701/http://suo.ieee.org/ontology/msg04070.html
- http://web.archive.org/web/20110612003540/http://suo.ieee.org/ontology/msg04072.html
- http://web.archive.org/web/20110612005229/http://suo.ieee.org/ontology/msg04073.html
- http://web.archive.org/web/20110610153117/http://suo.ieee.org/ontology/msg04074.html
- http://web.archive.org/web/20110612010555/http://suo.ieee.org/ontology/msg04077.html
- http://web.archive.org/web/20110612001918/http://suo.ieee.org/ontology/msg04079.html
- http://web.archive.org/web/20110612005244/http://suo.ieee.org/ontology/msg04080.html
- http://web.archive.org/web/20110612005249/http://suo.ieee.org/ontology/msg04268.html
- http://web.archive.org/web/20110612010626/http://suo.ieee.org/ontology/msg04269.html
- http://web.archive.org/web/20110612000853/http://suo.ieee.org/ontology/msg04272.html
- http://web.archive.org/web/20110612010514/http://suo.ieee.org/ontology/msg04273.html
- http://web.archive.org/web/20110612002235/http://suo.ieee.org/ontology/msg04290.html

#### DLOG A • Inquiry List

- http://stderr.org/pipermail/inquiry/2003-April/000372.html
- http://stderr.org/pipermail/inquiry/2003-April/000373.html
- http://stderr.org/pipermail/inquiry/2003-April/000374.html
- http://stderr.org/pipermail/inquiry/2003-April/000375.html
- http://stderr.org/pipermail/inquiry/2003-April/000376.html
- http://stderr.org/pipermail/inquiry/2003-April/000377.html
- http://stderr.org/pipermail/inquiry/2003-April/000378.html
- http://stderr.org/pipermail/inquiry/2003-April/000379.html
- http://stderr.org/pipermail/inquiry/2003-April/000380.html
- http://stderr.org/pipermail/inquiry/2003-April/000381.html
- http://stderr.org/pipermail/inquiry/2003-April/000382.html
- http://stderr.org/pipermail/inquiry/2003-April/000383.html
- http://stderr.org/pipermail/inquiry/2003-April/000384.html
- http://stderr.org/pipermail/inquiry/2003-April/000385.html
- http://stderr.org/pipermail/inquiry/2003-April/000386.html
- http://stderr.org/pipermail/inquiry/2003-April/000387.html
- http://stderr.org/pipermail/inquiry/2003-April/000388.html
- http://stderr.org/pipermail/inquiry/2003-April/000389.html
- http://stderr.org/pipermail/inquiry/2003-April/000390.html
- http://stderr.org/pipermail/inquiry/2003-April/000391.html
- http://stderr.org/pipermail/inquiry/2003-April/000392.html

### Differential Logic • Series B • History

#### DLOG B • Ontology List

- http://web.archive.org/web/20080725050148/http://suo.ieee.org/ontology/thrd47.html#03084
- http://web.archive.org/web/20070301065551/http://suo.ieee.org/ontology/thrd13.html#04798

- http://web.archive.org/web/20081204194718/http://suo.ieee.org/ontology/msg03084.html
- http://web.archive.org/web/20081204201329/http://suo.ieee.org/ontology/msg03092.html
- http://web.archive.org/web/20080907145125/http://suo.ieee.org/ontology/msg03093.html
- http://web.archive.org/web/20081204193400/http://suo.ieee.org/ontology/msg03095.html
- http://web.archive.org/web/20080725050452/http://suo.ieee.org/ontology/msg03100.html
- http://web.archive.org/web/20080807163836/http://suo.ieee.org/ontology/msg03109.html
- http://web.archive.org/web/20080807161500/http://suo.ieee.org/ontology/msg03110.html
- http://web.archive.org/web/20080807163330/http://suo.ieee.org/ontology/msg03112.html
- http://web.archive.org/web/20081012034633/http://suo.ieee.org/ontology/msg04798.html

#### DLOG B • Inquiry List

- http://stderr.org/pipermail/inquiry/2003-April/thread.html#393
- http://stderr.org/pipermail/inquiry/2003-May/thread.html#477

- http://stderr.org/pipermail/inquiry/2003-April/000393.html
- http://stderr.org/pipermail/inquiry/2003-April/000394.html
- http://stderr.org/pipermail/inquiry/2003-April/000397.html
- http://stderr.org/pipermail/inquiry/2003-April/000398.html
- http://stderr.org/pipermail/inquiry/2003-April/000399.html
- http://stderr.org/pipermail/inquiry/2003-April/000400.html
- http://stderr.org/pipermail/inquiry/2003-April/000401.html
- http://stderr.org/pipermail/inquiry/2003-April/000402.html
- http://stderr.org/pipermail/inquiry/2003-May/000477.html

### Differential Logic • Series C • History

#### DLOG C • Ontology List

#### DLOG C • Inquiry List

### Differential Logic • Series D • History

#### DLOG D • Ontology List

**Differential Logic and Dynamic Systems**

**1. Review and Transition**- D01. http://web.archive.org/web/20080725044922/http://suo.ieee.org/ontology/msg04799.html

**2. Functional Conception of Propositional Calculus**- D02. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04800.html
- D03. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04801.html
- D04. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04802.html
- D05. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04803.html
- D06. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04804.html
- D07. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04805.html
- D08. http://web.archive.org/web/20070302153557/http://suo.ieee.org/ontology/msg04806.html
- D09. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04807.html
- D10. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04808.html
- D11. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04809.html

**3. Differential Extension of Propositional Calculus**- D12. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04810.html
- D13. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04811.html
- D14. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04812.html
- D15. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04813.html
- D16. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04814.html
- D17. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04815.html

**4. Back to the Beginning : Some Exemplary Universes**- D18. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04816.html
- D19. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04817.html
- D20. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04818.html
- D21. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04819.html
- D22. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04820.html
- D23. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04821.html
- D24. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04822.html

**5. Transformations of Discourse**- D25. http://web.archive.org/web/20080906121910/http://suo.ieee.org/ontology/msg04823.html

**5.1. Foreshadowing Transformations : Extensions and Projections of Discourse**- D26. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04824.html
- D27. http://web.archive.org/web/20070304145312/http://suo.ieee.org/ontology/msg04825.html

**5.2. Thematization of Functions : And a Declaration of Independence for Variables**- D28. http://web.archive.org/web/20080905125010/http://suo.ieee.org/ontology/msg04826.html
- D29. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04827.html
- D30. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04828.html
- D31. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04829.html
- D32. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04830.html
- D33. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04832.html

**5.3. Propositional Transformations**- D34. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04833.html
- D35. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04834.html
- D36. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04835.html

**5.4. Analytic Expansions : Operators and Functors**- D37. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04836.html
- D38. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04837.html
- D39. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04838.html
- D40. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04839.html
- D41. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04840.html
- D42. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04841.html
- D43. http://web.archive.org/web/20070304025245/http://suo.ieee.org/ontology/msg04842.html
- D44. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04843.html
- D45. http://web.archive.org/web/20070303180640/http://suo.ieee.org/ontology/msg04844.html

**5.5. Transformations of Type B**^{2}→ B^{1}- D46. http://web.archive.org/web/20080906120746/http://suo.ieee.org/ontology/msg04845.html
- D47. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04846.html
- D48. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04847.html
- D49. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04848.html
- D50. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04849.html
- D51. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04850.html
- D52. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04851.html
- D53. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04852.html
- D54. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04853.html
- D55. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04854.html
- D56. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04855.html
- D57. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04856.html
- D58. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04857.html
- D59. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04858.html
- D60. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04859.html
- D61. http://web.archive.org/web/20070304145621/http://suo.ieee.org/ontology/msg04860.html
- D62. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04861.html
- D63. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04862.html
- D64. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04863.html
- D65. http://web.archive.org/web/20070304025432/http://suo.ieee.org/ontology/msg04864.html
- D66. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04865.html
- D67. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04866.html
- D68. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04867.html
- D69. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04868.html
- D70. http://web.archive.org/web/20070304212202/http://suo.ieee.org/ontology/msg04869.html

**5.6. Taking Aim at Higher Dimensional Targets**- D71. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04870.html
- D72. http://web.archive.org/web/20070304214116/http://suo.ieee.org/ontology/msg04871.html

**5.7. Transformations of Type B**^{2}→ B^{2}- D73. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04872.html
- D74. http://web.archive.org/web/20070304214126/http://suo.ieee.org/ontology/msg04873.html
- D75. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04874.html
- D76. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04875.html
- D77. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04876.html
- D78. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04877.html
- D79. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04878.html
- D80. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04879.html
- D81. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04880.html
- D82. http://web.archive.org/web/20070304205932/http://suo.ieee.org/ontology/msg04882.html
- D83. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04883.html

**Epilogue, Enchoiry, Exodus**- D84. http://web.archive.org/web/20081011050015/http://suo.ieee.org/ontology/msg04884.html

#### DLOG D • Inquiry List

**Differential Logic and Dynamic Systems**

**1. Review and Transition**

**2. A Functional Conception of Propositional Calculus**

**2.1. Qualitative Logic and Quantitative Analogy**

**2.2. Philosophy of Notation : Formal Terms and Flexible Types**

**2.3. Special Classes of Propositions**

**2.4. Basis Relativity and Type Ambiguity**

**2.5. The Analogy Between Real and Boolean Types**

**2.6. Theory of Control and Control of Theory**

**2.7. Propositions as Types and Higher Order Types**

**2.8. Reality at the Threshold of Logic**

**2.9. Tables of Propositional Forms**

**3. A Differential Extension of Propositional Calculus**

**3.1. Differential Propositions : The Qualitative Analogue of Differential Equations**

**3.2. An Interlude on the Path**

**3.3. The Extended Universe of Discourse**

**3.4. Intentional Propositions**

**3.5. Life on Easy Street**

**4. Back to the Beginning : Some Exemplary Universes**

**4.1. A One-Dimensional Universe**

**4.2. Example 1. A Square Rigging**

**4.3. Back to the Feature**

**4.4. Tacit Extensions**

**4.5. Example 2. Drives and Their Vicissitudes**

**5. Transformations of Discourse**

**5.1. Foreshadowing Transformations : Extensions and Projections of Discourse**

**5.1.1. Extension from 1 to 2 Dimensions**

**5.1.2. Extension from 2 to 4 Dimensions**

**5.2. Thematization of Functions : And a Declaration of Independence for Variables**

**5.2.1. Thematization : Venn Diagrams**

**5.2.2. Thematization : Truth Tables**

**5.3. Propositional Transformations**

**5.3.1. Alias and Alibi Transformations**

**5.3.2. Transformations of General Type**

**5.4. Analytic Expansions : Operators and Functors**

**5.4.1. Operators on Propositions and Transformations**

**5.4.2. Differential Analysis of Propositions and Transformations**

**5.4.2.1. The Secant Operator : $E$**

**5.4.2.2. The Radius Operator : $e$**

**5.4.2.3. The Phantom of the Operators : !h!**

**5.4.2.4. The Chord Operator : $D$**

**5.4.2.5. The Tangent Operator : $T$**

**5.5. Transformations of Type B**^{2}→ B^{1}

**5.5.1. Analytic Expansion of Conjunction**

**5.5.1.1. Tacit Extension of Conjunction**

**5.5.1.2. Enlargement Map of Conjunction**

**5.5.1.3. Digression : Reflection on Use and Mention**

**5.5.1.4. Difference Map of Conjunction**

**5.5.1.5. Differential of Conjunction**

**5.5.1.6. Remainder of Conjunction**

**5.5.1.7. Summary of Conjunction**

**5.5.2. Analytic Series : Coordinate Method**

**5.5.3. Analytic Series : Recap**

**5.5.4. Terminological Interlude**

**5.5.5. End of Perfunctory Chatter : Time to Roll the Clip!**

**5.6. Taking Aim at Higher Dimensional Targets**

**5.7. Transformations of Type B**^{2}→ B^{2}- 73. http://stderr.org/pipermail/inquiry/2003-June/000557.html
- 74. http://stderr.org/pipermail/inquiry/2003-June/000558.html
- 75. http://stderr.org/pipermail/inquiry/2003-June/000559.html
- 76. http://stderr.org/pipermail/inquiry/2003-June/000560.html
- 77. http://stderr.org/pipermail/inquiry/2003-June/000562.html
- 78. http://stderr.org/pipermail/inquiry/2003-June/000563.html
- 79. http://stderr.org/pipermail/inquiry/2003-June/000564.html
- 80. http://stderr.org/pipermail/inquiry/2003-June/000565.html
- 81. http://stderr.org/pipermail/inquiry/2003-June/000566.html
- 82. http://stderr.org/pipermail/inquiry/2003-June/000569.html
- 83. http://stderr.org/pipermail/inquiry/2003-June/000570.html

**Epilogue, Enchoiry, Exodus**

## Differential Logic 2003–2004 • Document History • Work Area

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Differential Logic 2004 -- Ontology List o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Differential Logic A 01. http://suo.ieee.org/ontology/msg05359.html 02. http://suo.ieee.org/ontology/msg05360.html 03. http://suo.ieee.org/ontology/msg05365.html 04. http://suo.ieee.org/ontology/msg05366.html 05. http://suo.ieee.org/ontology/msg05367.html 06. http://suo.ieee.org/ontology/msg05368.html 07. http://suo.ieee.org/ontology/msg05370.html 08. http://suo.ieee.org/ontology/msg05371.html 09. http://suo.ieee.org/ontology/msg05372.html 10. http://suo.ieee.org/ontology/msg05373.html 11. http://suo.ieee.org/ontology/msg05374.html 12. http://suo.ieee.org/ontology/msg05375.html 13. http://suo.ieee.org/ontology/msg05376.html 14. http://suo.ieee.org/ontology/msg05377.html 15. http://suo.ieee.org/ontology/msg05378.html 16. http://suo.ieee.org/ontology/msg05379.html 17. http://suo.ieee.org/ontology/msg05381.html 18. http://suo.ieee.org/ontology/msg05383.html 19. http://suo.ieee.org/ontology/msg05384.html 20. http://suo.ieee.org/ontology/msg05385.html 21. http://suo.ieee.org/ontology/msg05386.html DLOG. Differential Logic A -- Discussion 01. http://suo.ieee.org/ontology/msg05380.html 02. http://suo.ieee.org/ontology/msg05390.html 03. http://suo.ieee.org/ontology/msg05391.html 04. http://suo.ieee.org/ontology/msg05392.html 05. http://suo.ieee.org/ontology/msg05400.html 06. http://suo.ieee.org/ontology/msg05401.html 07. http://suo.ieee.org/ontology/msg05402.html 08. http://suo.ieee.org/ontology/msg05403.html 09. http://suo.ieee.org/ontology/msg05404.html 10. http://suo.ieee.org/ontology/msg05405.html 11. http://suo.ieee.org/ontology/msg05420.html 12. http://suo.ieee.org/ontology/msg05421.html 13. http://suo.ieee.org/ontology/msg05423.html 14. http://suo.ieee.org/ontology/msg05425.html 15. http://suo.ieee.org/ontology/msg05426.html 16. http://suo.ieee.org/ontology/msg05427.html 17. http://suo.ieee.org/ontology/msg05429.html 18. http://suo.ieee.org/ontology/msg05430.html 19. http://suo.ieee.org/ontology/msg05431.html 20. http://suo.ieee.org/ontology/msg05432.html 21. http://suo.ieee.org/ontology/msg05433.html 22. http://suo.ieee.org/ontology/msg05434.html 23. http://suo.ieee.org/ontology/msg05437.html 24. http://suo.ieee.org/ontology/msg05438.html 25. http://suo.ieee.org/ontology/msg05441.html DLOG. Differential Logic B 01. http://suo.ieee.org/ontology/msg05387.html 02. http://suo.ieee.org/ontology/msg05389.html 03. http://suo.ieee.org/ontology/msg05393.html 04. http://suo.ieee.org/ontology/msg05394.html 05. http://suo.ieee.org/ontology/msg05395.html 06. http://suo.ieee.org/ontology/msg05396.html 07. http://suo.ieee.org/ontology/msg05397.html 08. http://suo.ieee.org/ontology/msg05398.html 09. http://suo.ieee.org/ontology/msg05399.html 10. http://suo.ieee.org/ontology/msg05440.html 11. http://suo.ieee.org/ontology/msg05444.html 12. http://suo.ieee.org/ontology/msg05445.html 13. http://suo.ieee.org/ontology/msg05448.html 14. http://suo.ieee.org/ontology/msg05449.html 15. http://suo.ieee.org/ontology/msg05450.html 16. http://suo.ieee.org/ontology/msg05451.html 17. http://suo.ieee.org/ontology/msg05452.html 18. http://suo.ieee.org/ontology/msg05453.html 19. http://suo.ieee.org/ontology/msg05455.html 20. http://suo.ieee.org/ontology/msg05456.html DLOG. Differential Logic B -- Discussion 01. http://suo.ieee.org/ontology/msg05446.html 02. http://suo.ieee.org/ontology/msg05447.html 03. http://suo.ieee.org/ontology/msg05454.html DLOG. Differential Logic C 01. http://suo.ieee.org/ontology/msg05406.html o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Differential Logic 2004 -- Inquiry List o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Differential Logic A 00. http://stderr.org/pipermail/inquiry/2004-February/thread.html#1132 01. http://stderr.org/pipermail/inquiry/2004-February/001132.html 02. http://stderr.org/pipermail/inquiry/2004-February/001133.html 03. http://stderr.org/pipermail/inquiry/2004-February/001138.html 04. http://stderr.org/pipermail/inquiry/2004-February/001139.html 05. http://stderr.org/pipermail/inquiry/2004-February/001140.html 06. http://stderr.org/pipermail/inquiry/2004-February/001141.html 07. http://stderr.org/pipermail/inquiry/2004-February/001143.html 08. http://stderr.org/pipermail/inquiry/2004-February/001144.html 09. http://stderr.org/pipermail/inquiry/2004-February/001145.html 10. http://stderr.org/pipermail/inquiry/2004-February/001146.html 11. http://stderr.org/pipermail/inquiry/2004-February/001147.html 12. http://stderr.org/pipermail/inquiry/2004-February/001148.html 13. http://stderr.org/pipermail/inquiry/2004-February/001149.html 14. http://stderr.org/pipermail/inquiry/2004-February/001150.html 15. http://stderr.org/pipermail/inquiry/2004-February/001151.html 16. http://stderr.org/pipermail/inquiry/2004-February/001152.html 17. http://stderr.org/pipermail/inquiry/2004-February/001154.html 18. http://stderr.org/pipermail/inquiry/2004-February/001156.html 19. http://stderr.org/pipermail/inquiry/2004-February/001157.html 20. http://stderr.org/pipermail/inquiry/2004-February/001158.html 21. http://stderr.org/pipermail/inquiry/2004-February/001159.html DLOG. Differential Logic B 00. http://stderr.org/pipermail/inquiry/2004-February/thread.html#1160 01. http://stderr.org/pipermail/inquiry/2004-February/001160.html 02. http://stderr.org/pipermail/inquiry/2004-February/001161.html 03. http://stderr.org/pipermail/inquiry/2004-February/001165.html 04. http://stderr.org/pipermail/inquiry/2004-February/001166.html 05. http://stderr.org/pipermail/inquiry/2004-February/001167.html 06. http://stderr.org/pipermail/inquiry/2004-February/001168.html 07. http://stderr.org/pipermail/inquiry/2004-February/001169.html 08. http://stderr.org/pipermail/inquiry/2004-February/001170.html 09. http://stderr.org/pipermail/inquiry/2004-February/001171.html 10. http://stderr.org/pipermail/inquiry/2004-February/001209.html 11. http://stderr.org/pipermail/inquiry/2004-February/001213.html 12. http://stderr.org/pipermail/inquiry/2004-February/001214.html 13. http://stderr.org/pipermail/inquiry/2004-February/001217.html 14. http://stderr.org/pipermail/inquiry/2004-February/001218.html 15. http://stderr.org/pipermail/inquiry/2004-February/001219.html 16. http://stderr.org/pipermail/inquiry/2004-February/001220.html 17. http://stderr.org/pipermail/inquiry/2004-February/001221.html 18. http://stderr.org/pipermail/inquiry/2004-February/001222.html 19. http://stderr.org/pipermail/inquiry/2004-February/001225.html 20. http://stderr.org/pipermail/inquiry/2004-February/001226.html DLOG. Differential Logic C 00. http://stderr.org/pipermail/inquiry/2004-February/thread.html#1178 01. http://stderr.org/pipermail/inquiry/2004-February/001178.html DLOG. Differential Logic D 00. http://stderr.org/pipermail/inquiry/2003-May/thread.html#478 00. http://stderr.org/pipermail/inquiry/2003-June/thread.html#553 00. http://stderr.org/pipermail/inquiry/2003-June/thread.html#571 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DLOG. Differential Logic A -- Discussion 00. http://stderr.org/pipermail/inquiry/2004-February/thread.html#1153 01. http://stderr.org/pipermail/inquiry/2004-February/001153.html 02. http://stderr.org/pipermail/inquiry/2004-February/001162.html 03. http://stderr.org/pipermail/inquiry/2004-February/001163.html 04. http://stderr.org/pipermail/inquiry/2004-February/001164.html 05. http://stderr.org/pipermail/inquiry/2004-February/001172.html 06. http://stderr.org/pipermail/inquiry/2004-February/001173.html 07. http://stderr.org/pipermail/inquiry/2004-February/001174.html 08. http://stderr.org/pipermail/inquiry/2004-February/001175.html 09. http://stderr.org/pipermail/inquiry/2004-February/001176.html 10. http://stderr.org/pipermail/inquiry/2004-February/001177.html 11. http://stderr.org/pipermail/inquiry/2004-February/001192.html 12. http://stderr.org/pipermail/inquiry/2004-February/001193.html 13. http://stderr.org/pipermail/inquiry/2004-February/001194.html 14. http://stderr.org/pipermail/inquiry/2004-February/001195.html 15. http://stderr.org/pipermail/inquiry/2004-February/001196.html 16. http://stderr.org/pipermail/inquiry/2004-February/001197.html 17. http://stderr.org/pipermail/inquiry/2004-February/001198.html 18. http://stderr.org/pipermail/inquiry/2004-February/001199.html 19. http://stderr.org/pipermail/inquiry/2004-February/001200.html 20. http://stderr.org/pipermail/inquiry/2004-February/001201.html 21. http://stderr.org/pipermail/inquiry/2004-February/001202.html 22. http://stderr.org/pipermail/inquiry/2004-February/001203.html 23. http://stderr.org/pipermail/inquiry/2004-February/001206.html 24. http://stderr.org/pipermail/inquiry/2004-February/001207.html 25. http://stderr.org/pipermail/inquiry/2004-February/001210.html DLOG. Differential Logic B -- Discussion 00. http://stderr.org/pipermail/inquiry/2004-February/thread.html#1215 01. http://stderr.org/pipermail/inquiry/2004-February/001215.html 02. http://stderr.org/pipermail/inquiry/2004-February/001216.html 03. http://stderr.org/pipermail/inquiry/2004-February/001223.html 04. http://stderr.org/pipermail/inquiry/2004-February/001224.html 05. http://stderr.org/pipermail/inquiry/2004-February/001227.html 06. http://stderr.org/pipermail/inquiry/2004-February/001229.html o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o