[Inquiry] Re: Differential Logic

[Jon Awbrey]

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DLOG.  Note D14

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An Interlude on the Path

| There would have been no beginnings:
| instead, speech would proceed from me,
| while I stood in its path - a slender gap -
| the point of its possible disappearance.
|
| Michel Foucault, 'The Discourse on Language', [Fou, 215]

It may help to get a sense of the relation between B and D by considering the
"path classifier" (or equivalence class of curves) approach to tangent vectors.
As if by reflex, the thought of physical motion makes us cross over to a universe
marked by the nominal character [!X!].  Given the boolean value system, a path in
the space X = <|!X!|> is a map q : B -> X.  In this case, the set of paths (B -> X)
is isomorphic to the cartesian square X^2 = X x X, or the set of ordered pairs from X.

We may analyze X^2 = {<u, v> : u, v in X} into two parts,
specifically, the pairs that lie on and off the diagonal:

   X^2  =  {<u, v> : u = v}  |_|  {<u, v> : u =/= v}.

In symbolic terms, this partition may be expressed as:

   X^2 ~=~ Diag(X) + 2 * Comb(X, 2),

where:

   Diag(X)  =  {<x, x> : x in X},

and where:

   Comb(X, k)  =  "X choose k"  =  {k-sets from X},

so that:

   Comb(X, 2)  =  {{u, v} : u, v in X}.

We can now use the features in d!X! = {dx_1, ... , dx_n} to classify the paths
of (B -> X) by way of the pairs in X^2.  If X ~=~ B^n then a path in X has the
form q : (B -> B^n) ~=~ B^n x B^n ~=~ B^2n ~=~ (B^2)^n.  Intuitively, we want
to map this (B^2)^n onto D^n by mapping each component B^2 onto a copy of D.
But in our current situation "D" is just a name we give, or an accidental
quality we attribute, to coefficient values in B when they are attached
to features in d!X!.

Therefore, define dx_i : X^2 -> B such that:

   dx_i (<u, v>)  =  (| x_i (u) , x_i (v) |)

                  =  x_i (u)  +  x_i (v)

                  =  x_i (v)  -  x_i(u).

NB.  In the above transcription, "(| ... , ... |)" is a "cactus lobe",
signifying "just one false", in this case among two boolean variables,
while "+" is boolean addition in the proper sense of addition in GF(2),
and thus equivalent to "-", in the sense of adding the additive inverse.

The above definition is equivalent to defining dx_i : (B -> X) -> B such that:

   dx_i (q)  =  (| x_i (q_0) , x_i (q_1) |)

             =  x_i (q_0)  +  x_i (q_1)

             =  x_i (q_1)  -  x_i (q_0),

where q_b = q(b), for each b in B.  Thus, the proposition dx_i
is true of the path q = <u, v> exactly if the terms of q, the
endpoints u and v, lie on different sides of the question x_i.

Now we can use the language of features in <|d!X!|>, indeed the whole calculus
of propositions in [d!X!], to classify paths and sets of paths.  In other words,
the paths can be taken as models of the propositions g : dX -> B.  For example,
the paths corresponding to Diag(X) fall under the description (dx_1)...(dx_n),
which says that nothing changes among the set of features {x_1, ..., x_n}.

Finally, a few words of explanation may be in order.  If this concept of a path
appears to be described in a roundabout fashion, it is because I am trying to
avoid using any assumption of vector space properties for the space X which
contains its range.  In many ways the treatment is still unsatisfactory,
but improvements will have to wait for the introduction of substitution
operators acting on singular propositions.

Jon Awbrey

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