[Inquiry] Re: Dynamics And Logic

[Jon Awbrey]

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DAL.  Note 2

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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.

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| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `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 ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `@` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| 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.

   Let !P! be the set of two values {(p), p} = {not-p, p} ~=~ B

   Let !Q! be the set of two values {(q), q} = {not-q, q} ~=~ B

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:

   The "difference" operator <capital Delta>, written here as D.

   The "enlargement" operator <capital Epsilon>, written here as E.

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

In order to describe the universe in which these operators operate,
it will be necessary to enlarge our original universe of discourse.

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

   EX = X x dX = !P! x !Q! x d!P! x d!Q!

   where

     X   =  !P! x !Q!

    dX   =  d!P! x d!Q!

   d!P!  =  {(dp), dp}

   d!Q!  =  {(dq), dq}

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.

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| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `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

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inquiry e-lab: http://stderr.org/pipermail/inquiry/
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