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A New Kind of Science: The NKS Forum (http://forum.wolframscience.com/index.php)
- Pure NKS (http://forum.wolframscience.com/forumdisplay.php?forumid=3)
-- Dynamics And Logic (http://forum.wolframscience.com/showthread.php?threadid=420)

Posted by Jon Awbrey on 05-05-2004 06:00 PM:

Dynamics And Logic

Note (June 2009). The current version of this material can be found at this location.


DAL. Note 1

I am going to excerpt some of my previous explorations
on differential logic and dynamic systems and bring them
to bear on the sorts of discrete dynamical themes that we
find of interest in the NKS Forum. This adaptation draws on
the "Cactus Rules", "Propositional Equation Reasoning Systems",
and "Reductions Among Relations" threads, and will in time be
applied to the "Differential Analytic Turing Automata" thread:

CR. http://forum.wolframscience.com/sho...hp?threadid=256
PERS. http://forum.wolframscience.com/sho...hp?threadid=297
RAR. http://forum.wolframscience.com/sho...hp?threadid=400
DATA. http://forum.wolframscience.com/sho...hp?threadid=228

One of the first things that you can do, once you have
a moderately functional calculus for boolean functions
or propositional logic, whatever you choose to call it,
is to start thinking about, and even start computing,
the differentials of these functions or propositions.

Let us start with a proposition of the form "p and q",
that is graphed as two labels attached to a root node:

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` p q ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `@` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` p and q ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o

Written as a string, this is just the concatenation "p q".

The proposition pq may be taken as a boolean function f<p, q>
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 pq. Imagine yourself standing
in a fixed cell of the corresponding venn diagram, say,
the cell where the proposition pq is true, as pictured:

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `o-----------o` `o-----------o` ` ` ` ` |
| ` ` ` ` / ` ` ` ` ` ` \ / ` ` ` ` ` ` \ ` ` ` ` |
| ` ` ` `/` ` ` ` ` ` ` `o` ` ` ` ` ` ` `\` ` ` ` |
| ` ` ` / ` ` ` ` ` ` ` /%\ ` ` ` ` ` ` ` \ ` ` ` |
| ` ` `/` ` ` ` ` ` ` `/%%%\` ` ` ` ` ` ` `\` ` ` |
| ` ` o ` ` ` ` ` ` ` o%%%%%o ` ` ` ` ` ` ` o ` ` |
| ` ` | ` ` ` ` ` ` ` |%%%%%| ` ` ` ` ` ` ` | ` ` |
| ` ` | ` ` ` ` ` ` ` |%%%%%| ` ` ` ` ` ` ` | ` ` |
| ` ` | ` ` ` P ` ` ` |%%%%%| ` ` ` Q ` ` ` | ` ` |
| ` ` | ` ` ` ` ` ` ` |%%%%%| ` ` ` ` ` ` ` | ` ` |
| ` ` | ` ` ` ` ` ` ` |%%%%%| ` ` ` ` ` ` ` | ` ` |
| ` ` o ` ` ` ` ` ` ` o%%%%%o ` ` ` ` ` ` ` o ` ` |
| ` ` `\` ` ` ` ` ` ` `\%%%/` ` ` ` ` ` ` `/` ` ` |
| ` ` ` \ ` ` ` ` ` ` ` \%/ ` ` ` ` ` ` ` / ` ` ` |
| ` ` ` `\` ` ` ` ` ` ` `o` ` ` ` ` ` ` `/` ` ` ` |
| ` ` ` ` \ ` ` ` ` ` ` / \ ` ` ` ` ` ` / ` ` ` ` |
| ` ` ` ` `o-----------o` `o-----------o` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o

Now ask yourself: What is the value of the
proposition pq at a distance of dp and dq
from the cell pq where you are standing?

Don't think about it -- just compute:

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` dp o` `o dq ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` / \ / \ ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `p o---@---o q` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
| ` ` ` ` ` ` ` ` (p, dp) (q, dq) ` ` ` ` ` ` ` ` |
o-------------------------------------------------o

To make future graphs easier to draw in Asciiland,
I'll use devices like @=@=@ and o=o=o to identify
several nodes into one, as in this next redrawing:

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` p `dp q `dq ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` o---o o---o ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `\` | | `/` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ | | / ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `\| |/` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` @=@ ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
| ` ` ` ` ` ` ` ` (p, dp) (q, dq) ` ` ` ` ` ` ` ` |
o-------------------------------------------------o

However you draw it, these expressions follow because the
expression p + dp, where the plus sign indicates addition
in B and thus corresponds to the exclusive-or in logic,
parses to a graph of the following form:

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` p ` `dp ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `o---o` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `@` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` (p, dp) ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o

Next question: What is the difference between the value of the
proposition pq "over there", at a remove of dp dq, and the value
of the proposition pq where you are, all expressed in the form of
a general formula, of course? Here is the appropriate formulation:

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` p `dp q `dq ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` o---o o---o ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `\` | | `/` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` \ | | / ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `\| |/` ` ` ` `p q` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` o=o-----------o ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\` ` ` ` ` `/` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` \ ` ` ` ` / ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `\` ` ` `/` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ ` ` / ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `@` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
| ` ` ` ` ` ` `((p, dp)(q, dq), p q)` ` ` ` ` ` ` |
o-------------------------------------------------o

There is one thing that I ought to mention at this point:
Computed over B, plus and minus are identical operations.
This will make the relation between the differential and
the integral parts of the appropriate calculus slightly
stranger than usual, but we will get into that later.

Last question, for now: What is the value of this expression
from your current standpoint, that is, evaluated at the point
where pq is true? Well, substituting 1 for p and 1 for q in
the graph amounts to erasing the labels "p" and "q", like so:

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `dp ` `dq ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` o---o o---o ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `\` | | `/` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` \ | | / ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `\| |/` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` o=o-----------o ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\` ` ` ` ` `/` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` \ ` ` ` ` / ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `\` ` ` `/` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ ` ` / ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `@` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
| ` ` ` ` ` ` `(( , dp)( , dq), ` `)` ` ` ` ` ` ` |
o-------------------------------------------------o

And this is equivalent to the following graph:

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` dp` `dq ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `o` `o` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `@` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ((dp) (dq)) ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o

Jon Awbrey

Posted by Jon Awbrey on 05-05-2004 09:50 PM:

Dynamics And Logic

DAL. Note 2

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

p and q --Diff--> dp or dq.

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `dp ` dq` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` o ` o ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `\ /` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` o ` ` ` ` |
| ` ` ` `p q` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` `--Diff-->` ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
| ` ` ` `p q ` ` ` ` --Diff--> ` ` ((dp) (dq)) ` `|
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 has been the
lack of a syntax that was up to handling the complexity of
the expressions that evolve.

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<p, q> = pq.

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `o-----------o` `o-----------o` ` ` ` ` |
| ` ` ` ` / ` ` ` ` ` ` \ / ` ` ` ` ` ` \ ` ` ` ` |
| ` ` ` `/` ` ` ` ` ` ` `o` ` ` ` ` ` ` `\` ` ` ` |
| ` ` ` / ` ` ` ` ` ` ` /%\ ` ` ` ` ` ` ` \ ` ` ` |
| ` ` `/` ` ` ` ` ` ` `/%%%\` ` ` ` ` ` ` `\` ` ` |
| ` ` o ` ` ` ` ` ` ` o%%%%%o ` ` ` ` ` ` ` o ` ` |
| ` ` | ` ` ` ` ` ` ` |%%%%%| ` ` ` ` ` ` ` | ` ` |
| ` ` | ` ` ` ` ` ` ` |%%%%%| ` ` ` ` ` ` ` | ` ` |
| ` ` | ` ` ` P ` ` ` |% F %| ` ` ` Q ` ` ` | ` ` |
| ` ` | ` ` ` ` ` ` ` |%%%%%| ` ` ` ` ` ` ` | ` ` |
| ` ` | ` ` ` ` ` ` ` |%%%%%| ` ` ` ` ` ` ` | ` ` |
| ` ` o ` ` ` ` ` ` ` o%%%%%o ` ` ` ` ` ` ` o ` ` |
| ` ` `\` ` ` ` ` ` ` `\%%%/` ` ` ` ` ` ` `/` ` ` |
| ` ` ` \ ` ` ` ` ` ` ` \%/ ` ` ` ` ` ` ` / ` ` ` |
| ` ` ` `\` ` ` ` ` ` ` `o` ` ` ` ` ` ` `/` ` ` ` |
| ` ` ` ` \ ` ` ` ` ` ` / \ ` ` ` ` ` ` / ` ` ` ` |
| ` ` ` ` `o-----------o` `o-----------o` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` p q ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `@` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
| f = ` ` ` ` ` ` ` ` ` p q ` ` ` ` ` ` ` ` ` ` ` |
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.


Then interpret the usual propositions about p, q
as functions of concrete type f : !P! x !Q! -> B.

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

The first couple of operators that we need to consider are
logical analogues of the pair that play a founding role
in the classical "finite difference calculus", namely:

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.

Starting out from the initial space X = !P! x !Q!, we
construct its (first order) "differential extension":

The interpretations of these new symbols can be diverse,
but the easiest interpretation for now is just to say
that "dp" means "change p" and "dq" means "change q".

Drawing a venn diagram for the differential extension EX = X x dX
requires four logical dimensions, !P!, !Q!, d!P!, d!Q!, but it is
possible to project a suggestion of what the differential features
dp and dq are about on the 2-dimensional base space X = !P! x !Q!
by drawing arrows that cross the boundaries of the basic circles
in the venn diagram for X, reading an arrow as dp if it crosses
the boundary between p and (p) in either direction and reading
an arrow as dq if it crosses the boundary between q and (q)
in either direction.

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `o-----------o` `o-----------o` ` ` ` ` |
| ` ` ` ` / ` ` ` ` ` ` \ / ` ` ` ` ` ` \ ` ` ` ` |
| ` ` ` `/` ` ` `p` ` ` `o` ` ` `q` ` ` `\` ` ` ` |
| ` ` ` / ` ` ` ` ` ` ` /%\ ` ` ` ` ` ` ` \ ` ` ` |
| ` ` `/` ` ` ` ` ` ` `/%%%\` ` ` ` ` ` ` `\` ` ` |
| ` ` o ` ` ` ` ` ` ` o%%%%%o ` ` ` ` ` ` ` o ` ` |
| ` ` | ` ` ` ` ` ` ` |%%%%%| ` ` ` ` ` ` ` | ` ` |
| ` ` | ` ` ` ` dq ` `|%%%%%| ` `dp ` ` ` ` | ` ` |
| ` ` | ` ` <---------|--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 example, taking the differential proposition (dp (dq))
as saying that dp implies dq, in other words, that there
is "no change in p without a change in q".

Jon Awbrey

Posted by Jon Awbrey on 05-06-2004 04:06 AM:

Dynamics And Logic

DAL. Note 3

Given the proposition f<p, q> over the space X = !P! x !Q!,
the (first order) "enlargement" of f is the proposition Ef
over the differential extension EX that is defined by the
following formula:


In the example f<p, q> = pq, the enlargement Ef is given by:

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` p `dp q `dq ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` o---o o---o ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `\` | | `/` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ | | / ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `\| |/` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` @=@ ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
| Ef =` ` ` ` ` ` (p, dp) (q, dq) ` ` ` ` ` ` ` ` |
o-------------------------------------------------o

Given the proposition f<p, q> over X = !P! x !Q!, the
(first order) "difference" of f is the proposition Df
over EX that is defined by the formula Df = Ef - f, or,
written out in full:

In the example f<p, q> = pq, the difference Df is given by:

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` p `dp q `dq ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` o---o o---o ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `\ `| | `/` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` \ | | / ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `\| |/` ` ` ` `p q` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` o=o-----------o ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\` ` ` ` ` `/` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` \ ` ` ` ` / ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `\` ` ` `/` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ ` ` / ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `@` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
| Df =` ` ` ` ` `((p, dp)(q, dq), pq) ` ` ` ` ` ` |
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 pq,
in as much as if to say at the place where p = 1 and q = 1.
This evaluation is written in the form Df|pq or Df|<1, 1>,
and we arrived at the locally applicable law that states
that f = pq = p and q => Df|pq = ((dp)(dq)) = dp or dq.

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `o-----------o` `o-----------o` ` ` ` ` |
| ` ` ` ` / ` ` ` ` ` ` \ / ` ` ` ` ` ` \ ` ` ` ` |
| ` ` ` `/` ` ` `p` ` ` `o` ` ` `q` ` ` `\` ` ` ` |
| ` ` ` / ` ` ` ` ` ` ` /%\ ` ` ` ` ` ` ` \ ` ` ` |
| ` ` `/` ` ` ` ` ` ` `/%%%\` ` ` ` ` ` ` `\` ` ` |
| ` ` o ` ` ` ` ` ` ` o%%%%%o ` ` ` ` ` ` ` o ` ` |
| ` ` | ` ` ` ` ` ` ` |%%%%%| ` ` ` ` ` ` ` | ` ` |
| ` ` | ` ` `dq (dp)` |%%%%%| `dp (dq)` ` ` | ` ` |
| ` ` | ` o<----------|--o--|---------->o ` | ` ` |
| ` ` | ` ` ` ` ` ` ` |%%|%%| ` ` ` ` ` ` ` | ` ` |
| ` ` | ` ` ` ` ` ` ` |%%|%%| ` ` ` ` ` ` ` | ` ` |
| ` ` o ` ` ` ` ` ` ` o%%|%%o ` ` ` ` ` ` ` o ` ` |
| ` ` `\` ` ` ` ` ` ` `\%|%/` ` ` ` ` ` ` `/` ` ` |
| ` ` ` \ ` ` ` ` ` ` ` \|/ ` ` ` ` ` ` ` / ` ` ` |
| ` ` ` `\` ` ` ` ` ` ` `|` ` ` ` ` ` ` `/` ` ` ` |
| ` ` ` ` \ ` ` ` ` ` ` /|\ ` ` ` ` ` ` / ` ` ` ` |
| ` ` ` ` `o-----------o | o-----------o` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `dp|dq` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `v` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` dp` `dq ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `o` `o` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `@` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
| Df|pq = ` ` ` ` ` ((dp) (dq)) ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o

The picture illustrates the analysis of the inclusive
disjunction ((dp)(dq)) into the exclusive disjunction:
dp(dq) + (dp)dq + dp dq, a differential proposition that
may be interpreted to say "change p or change q 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 require
a complete and detailed description of ways to escape it.

Jon Awbrey

Posted by Jon Awbrey on 05-06-2004 03:30 PM:

Dynamics And Logic

DAL. Note 4

Last time we computed what will variously be called
the "difference map", the "difference proposition",
or the "local proposition" Df_x for the proposition
f<p, q> = pq at the point x where p = 1 and q = 1.

In the universe X = !P! x !Q!, the four propositions
pq, p(q), (p)q, (p)(q) 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_x = Df|x = Df|<1, 1> = Df|pq,
so long as we know the frame of reference in force.

Sticking with the example f<p, q> = pq, let us compute the
value of the difference proposition Df at all of the points.

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` p `dp q `dq ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` o---o o---o ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `\` | | `/` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` \ | | / ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `\| |/` ` ` ` `p q` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` o=o-----------o ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\` ` ` ` ` `/` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` \ ` ` ` ` / ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `\` ` ` `/` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ ` ` / ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `@` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
| Df =` ` ` ` ((p, dp)(q, dq), pq)` ` ` ` ` ` ` ` |
o-------------------------------------------------o

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `dp ` `dq ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` o---o o---o ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `\` | | `/` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` \ | | / ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `\| |/` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` o=o-----------o ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\` ` ` ` ` `/` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` \ ` ` ` ` / ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `\` ` ` `/` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ ` ` / ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `@` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
| Df|pq = ` ` ` ` ` ((dp) (dq)) ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `dp | `dq ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` o---o o---o ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `\` | | `/` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` \ | | / ` ` ` ` o ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `\| |/` ` ` ` ` | ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` o=o-----------o ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\` ` ` ` ` `/` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` \ ` ` ` ` / ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `\` ` ` `/` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ ` ` / ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `@` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
| Df|p(q) = ` ` ` ` `(dp) dq` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` | `dp ` `dq ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` o---o o---o ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `\` | | `/` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` \ | | / ` ` ` ` o ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `\| |/` ` ` ` ` | ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` o=o-----------o ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\` ` ` ` ` `/` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` \ ` ` ` ` / ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `\` ` ` `/` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ ` ` / ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `@` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
| Df|(p)q = ` ` ` ` ` `dp (dq)` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` o ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` | `dp | `dq ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` o---o o---o ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `\` | | `/` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` \ | | / ` ` ` o ` o ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `\| |/` ` ` ` `\ /` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` o=o-----------o ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\` ` ` ` ` `/` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` \ ` ` ` ` / ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `\` ` ` `/` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ ` ` / ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `@` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
| Df|(p)(q) = ` ` ` ` `dp dq` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o

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

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| `e` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| `o-o-o-...-o-o-o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` \ ` ` ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` `\` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` \ ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` `\` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` e ` ` ` ` |
| ` ` ` \ ` / ` ` ` ` ` ` ` ` ` ` ` ` ` o ` ` ` ` |
| ` ` ` `\ /` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` ` ` `=` ` ` ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
| `(e, , ... , , )` ` ` `=` ` ` ` ` ` `(e)` ` ` ` |
o-------------------------------------------------o

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| e_1 e_2 ` e_k `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| `o---o-...-o---o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` \ ` ` ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` `\` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` \ ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` `\` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` \ ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` `\ /` ` ` ` ` ` ` ` ` ` ` `e_1 ... e_k` ` |
| ` ` ` ` @ ` ` ` ` ` ` `=` ` ` ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
| `(e_1, ..., e_k, ())` `=` ` ` ` `e_1 ... e_k` ` |
o-------------------------------------------------o

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

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `dp|dq` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `o-----------o | o-----------o` ` ` ` ` |
| ` ` ` ` / ` ` ` ` ` ` \|/ ` ` ` ` ` ` \ ` ` ` ` |
| ` ` ` `/` ` ` `p` ` ` `|` ` ` `q` ` ` `\` ` ` ` |
| ` ` ` / ` ` ` ` ` ` ` /|\ ` ` ` ` ` ` ` \ ` ` ` |
| ` ` `/` ` ` ` ` ` ` `/%|%\` ` ` ` ` ` ` `\` ` ` |
| ` ` o ` ` ` ` ` ` ` o%%|%%o ` ` ` ` ` ` ` o ` ` |
| ` ` | ` ` `(dp) dq` |%%v%%| `dp (dq)` ` ` | ` ` |
| ` ` | ` o-----------|->o<-|-----------o ` | ` ` |
| ` ` | ` ` ` ` ` ` ` |%%%%%| ` ` ` ` ` ` ` | ` ` |
| ` ` | ` o<----------|--o--|---------->o ` | ` ` |
| ` ` | ` ` `(dp) dq` |%%|%%| `dp (dq)` ` ` | ` ` |
| ` ` o ` ` ` ` ` ` ` o%%|%%o ` ` ` ` ` ` ` o ` ` |
| ` ` `\` ` ` ` ` ` ` `\%|%/` ` ` ` ` ` ` `/` ` ` |
| ` ` ` \ ` ` ` ` ` ` ` \|/ ` ` ` ` ` ` ` / ` ` ` |
| ` ` ` `\` ` ` ` ` ` ` `|` ` ` ` ` ` ` `/` ` ` ` |
| ` ` ` ` \ ` ` ` ` ` ` /|\ ` ` ` ` ` ` / ` ` ` ` |
| ` ` ` ` `o-----------o | o-----------o` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `dp|dq` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `v` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o

This just amounts to a depiction of the points,
truth-value assignments, or interpretations in
EX = !P! x !Q! x d!P! x d!Q! that are indicated
by the difference map Df : EX -> B, namely, the
following six points or singular propositions:


By inspection, it is fairly easy to understand Df
as telling you what you have to do from each point
of X in order to change the value borne by f<p, q>
at the point in question, that is, in order to get
to a point where the value of f<p, q> is different
from what it is where you started.

Jon Awbrey

Posted by Jon Awbrey on 05-06-2004 06:00 PM:

Dynamics And Logic

DAL. Note 5

We have been studying the action of the difference operator D,
also known as the "localization operator", on the proposition
f : !P! x !Q! -> B that is commonly called the conjunction pq.
We categorized Df as a (first order) differential proposition,
a proposition of the type Df : !P! x !Q! x d!P! x d!Q! -> B.

Abstracting from the augmented venn diagram that shows how the
models or the satisfying interpretations of Df distribute over
the (first order) extended space EX = !P! x !Q! x d!P! x d!Q!,
we can represent Df in the form of a digraph or directed graph,
one whose points are labeled with the elements of X = !P! x !Q!
and whose arcs are labeled with the elements of dX = d!P! x d!Q!.

o-------------------------------------------------o
| `f =` ` ` ` ` ` ` ` ` p q ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| Df =` ` ` ` ` ` ` p `q` ((dp)(dq))` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` + ` ` ` p (q) `(dp) dq` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` + ` ` `(p) q` ` dp (dq) ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` + ` ` `(p)(q) ` dp` dq` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` p q ` ` ` ` ` ` ` ` ` ` ` |
| `p (q) o<------------->o<------------->o (p) q` |
| ` ` ` ` ` ` (dp) dq ` `^` ` dp (dq) ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` dp | dq ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `v` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` (p) (q) ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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.

Jon Awbrey

Posted by Jon Awbrey on 05-06-2004 08:00 PM:

Dynamics And Logic

DAL. Note 6

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<p, q> = pq.

To begin we need to formulate a suitably generic
definition of the extended universe of discourse:


For a proposition f : X_1 x ... x X_k -> B,
the (first order) "enlargement" of f is the
proposition Ef : EX -> B that is defined by:

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

In the case of the conjunction f<p, q> = pq,
the enlargement Ef is formulated as follows:

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:

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 initial domain of discourse X = !P! x !Q!.

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` p `dp q `dq ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` o---o o---o ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `\` | | `/` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ | | / ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `\| |/` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` @=@ ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
| Ef =` ` ` ` ` ` (p, dp) (q, dq) ` ` ` ` ` ` ` ` |
o-------------------------------------------------o

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `dp ` `dq ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` o---o o---o ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `\` | | `/` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ | | / ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `\| |/` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` @=@ ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
| Ef|pq = ` ` ` ` ` `(dp) (dq)` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `dp | `dq ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` o---o o---o ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `\` | | `/` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ | | / ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `\| |/` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` @=@ ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
| Ef|p(q) = ` ` ` ` `(dp) `dq ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | `dp ` `dq ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` o---o o---o ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `\` | | `/` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ | | / ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `\| |/` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` @=@ ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
| Ef|(p)q = ` ` ` ` ` dp` (dq)` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` o ` ` o ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | `dp | `dq ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` o---o o---o ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `\` | | `/` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ | | / ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `\| |/` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` @=@ ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
| Ef|(p)(q) = ` ` ` ` dp` `dq ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o

Given the kind 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.

Here is a summary of the result, illustrated by means of
a digraph picture, where the "no change" element (dp)(dq)
is drawn as a loop at the point p q.

o-------------------------------------------------o
| `f =` ` ` ` ` ` ` ` ` p q ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| Ef =` ` ` ` ` ` ` p `q` `(dp)(dq) ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` + ` ` ` p (q) `(dp) dq` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` + ` ` `(p) q` ` dp (dq) ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` + ` ` `(p)(q) ` dp` dq` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `(dp) (dq)` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `.--->---.` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ ` ` / ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `\p q/` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` |
| `p (q) o-------------->o<--------------o (p) q` |
| ` ` ` ` ` ` (dp) dq ` `^` ` dp (dq) ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` dp | dq ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` (p) (q) ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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 in the universe X.

Jon Awbrey

Posted by Jon Awbrey on 05-06-2004 09:56 PM:

Dynamics And Logic

DAL. Note 7

To broaden our experience with simple examples, let us
now contemplate the sixteen functions of concrete type
!P! x !Q! -> B and abstract type B x B -> B. For ease
of future reference, I will set here a few tables that
specify the actions of E and D on the 16 functions and
allow us to view the results in several different ways.

By way of initial orientation, Table 7 lists equivalent expressions
for the sixteen functions in several different formalisms, indexing
systems, or languages for the propositional calculus, also known as
"zeroth order logic" (ZOL).

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
| ` ` ` ` | ` ` ` p : 1 1 0 0 | ` ` ` ` `| ` ` ` ` ` ` ` ` `| ` ` ` ` `|
| ` ` ` ` | ` ` ` q : 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 | `(p)(q) `| neither p nor q `| ~p & ~q `|
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` `| ` ` ` ` ` ` ` ` `| ` ` ` ` `|
| f_2 ` ` | f_0010` | 0 0 1 0 | `(p) q` `| q and not p ` ` `| ~p & `q `|
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` `| ` ` ` ` ` ` ` ` `| ` ` ` ` `|
| f_3 ` ` | f_0011` | 0 0 1 1 | `(p)` ` `| not p ` ` ` ` ` `| ~p` ` ` `|
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` `| ` ` ` ` ` ` ` ` `| ` ` ` ` `|
| f_4 ` ` | f_0100` | 0 1 0 0 | ` p (q) `| p and not q ` ` `| `p & ~q `|
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` `| ` ` ` ` ` ` ` ` `| ` ` ` ` `|
| f_5 ` ` | f_0101` | 0 1 0 1 | ` ` (q) `| not q ` ` ` ` ` `| ` ` `~q `|
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` `| ` ` ` ` ` ` ` ` `| ` ` ` ` `|
| f_6 ` ` | f_0110` | 0 1 1 0 | `(p, q) `| p not equal to q | `p + `q `|
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` `| ` ` ` ` ` ` ` ` `| ` ` ` ` `|
| f_7 ` ` | f_0111` | 0 1 1 1 | `(p `q) `| not both p and q | ~p v ~q `|
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` `| ` ` ` ` ` ` ` ` `| ` ` ` ` `|
| f_8 ` ` | f_1000` | 1 0 0 0 | ` p `q` `| p and q ` ` ` ` `| `p & `q `|
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` `| ` ` ` ` ` ` ` ` `| ` ` ` ` `|
| f_9 ` ` | f_1001` | 1 0 0 1 | ((p, q)) | p equal to q` ` `| `p = `q `|
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` `| ` ` ` ` ` ` ` ` `| ` ` ` ` `|
| f_10` ` | f_1010` | 1 0 1 0 | ` ` `q` `| q ` ` ` ` ` ` ` `| ` ` ` q `|
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` `| ` ` ` ` ` ` ` ` `| ` ` ` ` `|
| f_11` ` | f_1011` | 1 0 1 1 | `(p (q)) | not p without q `| `p => q `|
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` `| ` ` ` ` ` ` ` ` `| ` ` ` ` `|
| f_12` ` | f_1100` | 1 1 0 0 | ` p ` ` `| p ` ` ` ` ` ` ` `| `p` ` ` `|
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` `| ` ` ` ` ` ` ` ` `| ` ` ` ` `|
| f_13` ` | f_1101` | 1 1 0 1 | ((p) q) `| not q without p `| `p <= q `|
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` `| ` ` ` ` ` ` ` ` `| ` ` ` ` `|
| f_14` ` | f_1110` | 1 1 1 0 | ((p)(q)) | p or q` ` ` ` ` `| `p v `q `|
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` `| ` ` ` ` ` ` ` ` `| ` ` ` ` `|
| f_15` ` | f_1111` | 1 1 1 1 | ` (())` `| true` ` ` ` ` ` `| ` `1` ` `|
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` `| ` ` ` ` ` ` ` ` `| ` ` ` ` `|
o---------o---------o---------o----------o------------------o----------o

Jon Awbrey

Posted by Jon Awbrey on 05-07-2004 03:00 AM:

Dynamics And Logic

DAL. Note 8

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,
partitioned into seven natural classes by a group action.

Table 8-a. Ef Expanded Over Ordinary Features
o------o------------o------------o------------o------------o------------o
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
| ` ` `| ` ` f ` ` `| `Ef | pq` `| Ef | p(q) `| Ef | (p)q `| Ef | (p)(q)|
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
o------o------------o------------o------------o------------o------------o
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
| f_0 `| ` ` ()` ` `| ` ` ()` ` `| ` ` ()` ` `| ` ` ()` ` `| ` ` ()` ` `|
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
o------o------------o------------o------------o------------o------------o
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
| f_1 `| ` (p)(q)` `| ` dp` dq` `| ` dp (dq) `| `(dp) dq` `| `(dp)(dq) `|
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
| f_2 `| ` (p) q ` `| ` dp (dq) `| ` dp` dq` `| `(dp)(dq) `| `(dp) dq` `|
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
| f_4 `| ` `p (q)` `| `(dp) dq` `| `(dp)(dq) `| ` dp` dq` `| ` dp (dq) `|
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
| f_8 `| ` `p` q ` `| `(dp)(dq) `| `(dp) dq` `| ` dp (dq) `| ` dp` dq` `|
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
o------o------------o------------o------------o------------o------------o
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
| f_3 `| ` (p) ` ` `| ` dp` ` ` `| ` dp` ` ` `| `(dp) ` ` `| `(dp) ` ` `|
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
| f_12 | ` `p` ` ` `| `(dp) ` ` `| `(dp) ` ` `| ` dp` ` ` `| ` dp` ` ` `|
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
o------o------------o------------o------------o------------o------------o
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
| f_6 `| ` (p, q)` `| `(dp, dq) `| ((dp, dq)) | ((dp, dq)) | `(dp, dq) `|
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
| f_9 `| `((p, q)) `| ((dp, dq)) | `(dp, dq) `| `(dp, dq) `| ((dp, dq)) |
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
o------o------------o------------o------------o------------o------------o
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
| f_5 `| ` ` `(q)` `| ` ` ` dq` `| ` ` `(dq) `| ` ` ` dq` `| ` ` `(dq) `|
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
| f_10 | ` ` ` q ` `| ` ` `(dq) `| ` ` ` dq` `| ` ` `(dq) `| ` ` ` dq` `|
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
o------o------------o------------o------------o------------o------------o
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
| f_7 `| ` (p` q)` `| ((dp)(dq)) | ((dp) dq) `| `(dp (dq)) | `(dp` dq) `|
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
| f_11 | ` (p (q)) `| ((dp) dq) `| ((dp)(dq)) | `(dp` dq) `| `(dp (dq)) |
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
| f_13 | `((p) q)` `| `(dp (dq)) | `(dp` dq) `| ((dp)(dq)) | ((dp) dq) `|
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
| f_14 | `((p)(q)) `| `(dp` dq) `| `(dp (dq)) | ((dp) dq) `| ((dp)(dq)) |
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
o------o------------o------------o------------o------------o------------o
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
| f_15 | ` `(()) ` `| ` `(()) ` `| ` `(()) ` `| ` `(()) ` `| ` `(()) ` `|
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
o------o------------o------------o------------o------------o------------o

Table 8-b. Df Expanded Over Ordinary Features
o------o------------o------------o------------o------------o------------o
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
| ` ` `| ` ` f ` ` `| `Df | pq` `| Df | p(q) `| Df | (p)q `| Df | (p)(q)|
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
o------o------------o------------o------------o------------o------------o
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
| f_0 `| ` ` ()` ` `| ` ` ()` ` `| ` ` ()` ` `| ` ` ()` ` `| ` ` ()` ` `|
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
o------o------------o------------o------------o------------o------------o
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
| f_1 `| ` (p)(q)` `| ` dp` dq` `| ` dp (dq) `| `(dp) dq` `| ((dp)(dq)) |
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
| f_2 `| ` (p) q ` `| ` dp (dq) `| ` dp` dq` `| ((dp)(dq)) | `(dp) dq` `|
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
| f_4 `| ` `p (q)` `| `(dp) dq` `| ((dp)(dq)) | ` dp` dq` `| ` dp (dq) `|
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
| f_8 `| ` `p` q ` `| ((dp)(dq)) | `(dp) dq` `| ` dp (dq) `| ` dp` dq` `|
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
o------o------------o------------o------------o------------o------------o
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
| f_3 `| ` (p) ` ` `| ` dp` ` ` `| ` dp` ` ` `| ` dp` ` ` `| ` dp` ` ` `|
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
| f_12 | ` `p` ` ` `| ` dp` ` ` `| ` dp` ` ` `| ` dp` ` ` `| ` dp` ` ` `|
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
o------o------------o------------o------------o------------o------------o
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
| f_6 `| ` (p, q)` `| `(dp, dq) `| `(dp, dq) `| `(dp, dq) `| `(dp, dq) `|
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
| f_9 `| `((p, q)) `| `(dp, dq) `| `(dp, dq) `| `(dp, dq) `| `(dp, dq) `|
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
o------o------------o------------o------------o------------o------------o
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
| f_5 `| ` ` `(q)` `| ` ` ` dq` `| ` ` ` dq` `| ` ` ` dq` `| ` ` ` dq` `|
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
| f_10 | ` ` ` q ` `| ` ` ` dq` `| ` ` ` dq` `| ` ` ` dq` `| ` ` ` dq` `|
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
o------o------------o------------o------------o------------o------------o
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
| f_7 `| ` (p` q)` `| ((dp)(dq)) | `(dp) dq` `| ` dp (dq) `| ` dp` dq` `|
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
| f_11 | ` (p (q)) `| `(dp) dq` `| ((dp)(dq)) | ` dp` dq` `| ` dp (dq) `|
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
| f_13 | `((p) q)` `| ` dp (dq) `| ` dp` dq` `| ((dp)(dq)) | `(dp) dq` `|
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
| f_14 | `((p)(q)) `| ` dp` dq` `| ` dp (dq) `| `(dp) dq` `| ((dp)(dq)) |
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
o------o------------o------------o------------o------------o------------o
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
| f_15 | ` `(()) ` `| ` ` ()` ` `| ` ` ()` ` `| ` ` ()` ` `| ` ` ()` ` `|
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
o------o------------o------------o------------o------------o------------o

Jon Awbrey

Posted by Jon Awbrey on 05-07-2004 03:42 AM:

Dynamics And Logic

DAL. Note 9

Table 9-a. Ef Expanded Over Differential Features
o------o------------o------------o------------o------------o------------o
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
| ` ` `| ` ` f ` ` `| ` T_11 f` `| ` T_10 f` `| ` T_01 f` `| ` T_00 f` `|
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
| ` ` `| ` ` ` ` ` `| Ef| dp dq `| Ef| dp(dq) | Ef| (dp)dq | Ef|(dp)(dq)|
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
o------o------------o------------o------------o------------o------------o
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
| f_0 `| ` ` ()` ` `| ` ` ()` ` `| ` ` ()` ` `| ` ` ()` ` `| ` ` ()` ` `|
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
o------o------------o------------o------------o------------o------------o
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
| f_1 `| ` (p)(q)` `| ` `p` q ` `| ` `p (q)` `| ` (p) q ` `| ` (p)(q)` `|
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
| f_2 `| ` (p) q ` `| ` `p (q)` `| ` `p` q ` `| ` (p)(q)` `| ` (p) q ` `|
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
| f_4 `| ` `p (q)` `| ` (p) q ` `| ` (p)(q)` `| ` `p` q ` `| ` `p (q)` `|
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
| f_8 `| ` `p` q ` `| ` (p)(q)` `| ` (p) q ` `| ` `p (q)` `| ` `p` q ` `|
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
o------o------------o------------o------------o------------o------------o
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
| f_3 `| ` (p) ` ` `| ` `p` ` ` `| ` `p ` ` ` | ` (p) ` ` `| ` (p) ` ` `|
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
| f_12 | ` `p` ` ` `| ` (p) ` ` `| ` (p) ` ` `| ` `p` ` ` `| ` `p` ` ` `|
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
o------o------------o------------o------------o------------o------------o
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
| f_6 `| ` (p, q)` `| ` (p, q)` `| `((p, q)) `| `((p, q)) `| ` (p, q)` `|
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
| f_9 `| `((p, q)) `| `((p, q)) `| ` (p, q)` `| ` (p, q)` `| `((p, q)) `|
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
o------o------------o------------o------------o------------o------------o
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
| f_5 `| ` ` `(q)` `| ` ` ` q ` `| ` ` `(q)` `| ` ` ` q ` `| ` ` `(q)` `|
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
| f_10 | ` ` ` q ` `| ` ` `(q)` `| ` ` ` q ` `| ` ` `(q)` `| ` ` ` q ` `|
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
o------o------------o------------o------------o------------o------------o
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
| f_7 `| ` (p` q)` `| `((p)(q)) `| `((p) q)` `| ` (p (q)) `| ` (p` q)` `|
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
| f_11 | ` (p (q)) `| `((p) q)` `| `((p)(q)) `| ` (p` q)` `| ` (p (q)) `|
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
| f_13 | `((p) q)` `| ` (p (q)) `| ` (p` q)` `| `((p)(q)) `| `((p) q)` `|
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
| f_14 | `((p)(q)) `| ` (p` q)` `| ` (p (q)) `| `((p) q)` `| `((p)(q)) `|
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
o------o------------o------------o------------o------------o------------o
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
| f_15 | ` `(()) ` `| ` `(()) ` `| ` `(()) ` `| ` `(()) ` `| ` `(()) ` `|
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
o------o------------o------------o------------o------------o------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
| Fiped Point Total | ` ` `4` ` `| ` ` `4` ` `| ` ` `4` ` `| ` ` 16` ` `|
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
o-------------------o------------o------------o------------o------------o

Table 9-b. Df Expanded Over Differential Features
o------o------------o------------o------------o------------o------------o
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
| ` ` `| ` ` f ` ` `| Df| dp dq `| Df| dp(dq) | Df| (dp)dq | Df|(dp)(dq)|
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
o------o------------o------------o------------o------------o------------o
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
| f_0 `| ` ` ()` ` `| ` ` ()` ` `| ` ` ()` ` `| ` ` ()` ` `| ` ` ()` ` `|
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
o------o------------o------------o------------o------------o------------o
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
| f_1 `| ` (p)(q)` `| `((p, q)) `| ` `(q)` ` `| ` `(p)` ` `| ` ` ()` ` `|
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
| f_2 `| ` (p) q ` `| ` (p, q)` `| ` ` q ` ` `| ` `(p)` ` `| ` ` ()` ` `|
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
| f_4 `| ` `p (q)` `| ` (p, q)` `| ` `(q)` ` `| ` ` p ` ` `| ` ` ()` ` `|
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
| f_8 `| ` `p` q ` `| `((p, q)) `| ` ` q ` ` `| ` ` p ` ` `| ` ` ()` ` `|
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
o------o------------o------------o------------o------------o------------o
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
| f_3 `| ` (p) ` ` `| ` `(()) ` `| ` `(()) ` `| ` ` ()` ` `| ` ` ()` ` `|
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
| f_12 | ` `p` ` ` `| ` `(()) ` `| ` `(()) ` `| ` ` ()` ` `| ` ` ()` ` `|
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
o------o------------o------------o------------o------------o------------o
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
| f_6 `| ` (p, q)` `| ` ` ()` ` `| ` `(()) ` `| ` `(()) ` `| ` ` ()` ` `|
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
| f_9 `| `((p, q)) `| ` ` ()` ` `| ` `(()) ` `| ` `(()) ` `| ` ` ()` ` `|
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
o------o------------o------------o------------o------------o------------o
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
| f_5 `| ` ` `(q)` `| ` `(()) ` `| ` ` ()` ` `| ` `(()) ` `| ` ` ()` ` `|
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
| f_10 | ` ` ` q ` `| ` `(()) ` `| ` ` ()` ` `| ` `(()) ` `| ` ` ()` ` `|
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
o------o------------o------------o------------o------------o------------o
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
| f_7 `| ` (p` q)` `| `((p, q)) `| ` ` q ` ` `| ` ` p ` ` `| ` ` ()` ` `|
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
| f_11 | ` (p (q)) `| ` (p, q)` `| ` `(q)` ` `| ` ` p ` ` `| ` ` ()` ` `|
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
| f_13 | `((p) q)` `| ` (p, q)` `| ` ` q ` ` `| ` `(p)` ` `| ` ` ()` ` `|
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
| f_14 | `((p)(q)) `| `((p, q)) `| ` `(q)` ` `| ` `(p)` ` `| ` ` ()` ` `|
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
o------o------------o------------o------------o------------o------------o
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
| f_15 | ` `(()) ` `| ` ` ()` ` `| ` ` ()` ` `| ` ` ()` ` `| ` ` ()` ` `|
| ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `|
o------o------------o------------o------------o------------o------------o

Jon Awbrey

Posted by Jon Awbrey on 05-07-2004 03:20 PM:

Dynamics And Logic

DAL. Note 10

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 need 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 9-a, above or here:

DAL 9. http://forum.wolframscience.com/sho...d=1301#post1301
DAL 9. http://stderr.org/pipermail/inquiry...May/001408.html

The shift operator E can be understood as enacting
a substitution operation on the proposition f that
is given as its argument. In our present focus on
propositional forms that involve two variables, we
have the following datatype and applied definition:


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

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 : X -> B into some other
proposition f' : X -> 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 linear and local
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!

Jon Awbrey

Posted by Jon Awbrey on 05-07-2004 06:08 PM:

Dynamics And Logic

DAL. Note 11

We have been contemplating functions of the type f : X -> 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 a pair of basic connectives in the cactus language for ZOL.

Consider the proposition f of concrete type f : !P! x !Q! x !R! -> B
and abstract type f : B^3 -> B that is written as "(p, q, r)" in the
cactus syntax. Taken as an assertion in what C.S. Peirce called the
"existential interpretation", the so-called boundary form "(p, q, r)"
asserts that one and only one of the propositions p, q, r is false.
It is instructive to consider this assertion in relation to the
conjunction "p q r" of the same propositions. A venn diagram
for the boundary form (p, q, r) is shown in Figure 11.

o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` X ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `o-------------o` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` / ` ` ` ` ` ` ` \ ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` `\` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` / ` ` ` ` ` ` ` ` ` \ ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` `\` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` / ` ` ` ` ` ` ` ` ` ` ` \ ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|` ` ` ` ` ` P ` ` ` ` ` `|` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` o--o----------o ` o----------o--o ` ` ` ` ` ` |
| ` ` ` ` ` `/` ` \%%%%%%%%%%\ /%%%%%%%%%%/ ` `\` ` ` ` ` ` |
| ` ` ` ` ` / ` ` `\%%%%%%%%%%o%%%%%%%%%%/` ` ` \ ` ` ` ` ` |
| ` ` ` ` `/` ` ` ` \%%%%%%%%/ \%%%%%%%%/ ` ` ` `\` ` ` ` ` |
| ` ` ` ` / ` ` ` ` `\%%%%%%/ ` \%%%%%%/` ` ` ` ` \ ` ` ` ` |
| ` ` ` `/` ` ` ` ` ` \%%%%/` ` `\%%%%/ ` ` ` ` ` `\` ` ` ` |
| ` ` ` o ` ` ` ` ` ` `o--o-------o--o` ` ` ` ` ` ` o ` ` ` |
| ` ` ` | ` ` ` ` ` ` ` ` |%%%%%%%| ` ` ` ` ` ` ` ` | ` ` ` |
| ` ` ` | ` ` ` ` ` ` ` ` |%%%%%%%| ` ` ` ` ` ` ` ` | ` ` ` |
| ` ` ` | ` ` ` ` ` ` ` ` |%%%%%%%| ` ` ` ` ` ` ` ` | ` ` ` |
| ` ` ` | ` ` ` `Q` ` ` ` |%%%%%%%| ` ` ` `R` ` ` ` | ` ` ` |
| ` ` ` | ` ` ` ` ` ` ` ` |%%%%%%%| ` ` ` ` ` ` ` ` | ` ` ` |
| ` ` ` o ` ` ` ` ` ` ` ` o%%%%%%%o ` ` ` ` ` ` ` ` o ` ` ` |
| ` ` ` `\` ` ` ` ` ` ` ` `\%%%%%/` ` ` ` ` ` ` ` `/` ` ` ` |
| ` ` ` ` \ ` ` ` ` ` ` ` ` \%%%/ ` ` ` ` ` ` ` ` / ` ` ` ` |
| ` ` ` ` `\` ` ` ` ` ` ` ` `\%/` ` ` ` ` ` ` ` `/` ` ` ` ` |
| ` ` ` ` ` \ ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` / ` ` ` ` ` |
| ` ` ` ` ` `\` ` ` ` ` ` ` `/ \` ` ` ` ` ` ` `/` ` ` ` ` ` |
| ` ` ` ` ` ` o-------------o ` o-------------o ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
Figure 11. Boundary Form (p, q, r)

In relation to the center cell indicated by the conjunction pqr
the region indicated by (p, q, r) 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, pqr.

More generally speaking, in a k-dimensional universe of discourse
that is based on the "alphabet" of features !X! = {x_1, ..., x_k},
the same form of boundary relationship is manifested for any cell
of origin that one might choose 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.

Jon Awbrey

Posted by Jon Awbrey on 05-07-2004 08:00 PM:

Dynamics And Logic

DAL. Note 12

| 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, "Maxim of Pragmaticism", 'Collected Papers', 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 took up
earlier. That is to say, the story is the same, only the names have
been changed. An abstract group can have a variety of significantly
and superficially different representations. But even after we have
long forgotten the details of any particular representation there is
a type of concrete representations, called "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:


And so, by expanding effects, we get:

More on the pragmatic maxim as a representation principle later.

Jon Awbrey

Posted by Jon Awbrey on 05-08-2004 02:28 AM:

Dynamics And Logic

DAL. Note 13

The above-mentioned fact about the regular representations
of a group is universally known as "Cayley's Theorem". It
is usually stated in the form: "Every group is isomorphic
to a subgroup of Aut(X), where X is a suitably chosen set
and Aut(X) is the group of its automorphisms". There is
in Peirce's early papers a considerable generalization
of the concept of regular representations to a broad
class of relational algebraic systems. The crux of
the whole idea can be summed up as follows:


This idea of definition by way of context transforming operators
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, 'From Frege to Gödel', p. 216).
Today we'd call these constructions "term models". This, again, is
the big idea behind Schönfinkel's combinators {S, K, I}, and hence
of lambda calculus, and I reckon you all know where that leads.

Jon Awbrey

Posted by Jon Awbrey on 05-08-2004 06:14 PM:

Dynamics And Logic

DAL. Note 14

The next few excursions in this series will provide
a scenic tour of various ideas in group theory that
will turn out to be of constant guidance in several
of the settings that are associated with our topic.

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.

Peirce is well aware that it is not at all necessary to arrange the
elementary relatives of a relation into arrays, matrices, or tables,
but when he does so he tends to prefer organizing 2-adic relations
in the following manner:


For example, given the set X = {a, b, c}, suppose that
we have the 2-adic relative term m = "marker for" and
the associated 2-adic relation M c X x X, the general
pattern of whose common structure is represented by
the following matrix:

It has long been customary to omit the implicit plus signs
in these matrical displays, but I have restored them here
simply as a way of separating terms in this blancophage
web format.

For at least a little while, I will make explicit
the distinction between a "relative term" like m
and a "relation" like M c X x X, but it is best
to think of both of these entities as involving
different applications of the same information,
and so we could just as easily write this form:

By way of making up a concrete example,
let us say that M is given as follows:

In sum, we have this matrix:

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

Jon Awbrey

Posted by Jon Awbrey on 05-09-2004 03:12 AM:

Dynamics And Logic

DAL. Note 15

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:


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

Recognizing !1! = a:a + b:b + c:c as the identity transformation,
the 2-adic relative term m = "marker for" can be represented as an
element !1! + a:b + b:c + c:a of the so-called "group ring", all of
which makes this element just 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 customarily read them in
logical contexts: i is the relate, j is the correlate, and in
our current example we reading i:j, or more exactly, m_ij = 1,
to say that i is a marker for 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:

Jon Awbrey
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