12,080
edits
Changes
Directory:Jon Awbrey/Papers/Differential Propositional Calculus (view source)
Revision as of 20:44, 17 April 2008
, 20:44, 17 April 2008add content by Jon Awbrey
Of course, the place-settings of convenience at different symposia may vary.
Of course, the place-settings of convenience at different symposia may vary.
</pre>
</pre>
==Differential Logic : Series B==
===Note 1===
<pre>
| 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.
</pre>
===Note 2===
<pre>
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.
</pre>
===Note 3===
<pre>
| 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.
</pre>
===Note 4===
<pre>
| 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.
</pre>
===Note 5===
<pre>
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.
</pre>
===Note 6===
<pre>
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
</pre>
===Note 7===
<pre>
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".
</pre>
===Note 8===
<pre>
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.
</pre>
===Note 9===
<pre>
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
</pre>
===Note 10===
<pre>
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.
</pre>
===Note 11===
<pre>
| "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.
</pre>
===Note 12===
<pre>
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".
</pre>
===Note 13===
<pre>
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.
</pre>
===Note 14===
<pre>
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.
</pre>
===Note 15===
<pre>
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
</pre>
===Note 16===
<pre>
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.
</pre>
===Note 17===
<pre>
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))
</pre>
===Note 18===
<pre>
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.
</pre>
===Note 19===
<pre>
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))
</pre>
===Note 20===
<pre>
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.
</pre>
==Differential Logic : Series C==
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.
<pre>
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
</pre>
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.
<pre>
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
</pre>
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.
<pre>
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
</pre>
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.