ONT Re: Differential Logic
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DLOG. Note D13
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Differential Propositions: The Qualitative Analogue of Differential Equations
In order to define the differential extension of a universe of discourse [!A!],
the initial alphabet !A! must be extended to include a collection of symbols for
"differential features", or basic "changes" that are capable of occurring in [!A!].
Intuitively, these symbols may be construed as denoting primitive features of change,
qualitative attributes of motion, or propositions about how things or points in [!A!]
may change or move with respect to the features that are noted in the initial alphabet.
Hence, let us define the corresponding "differential alphabet" or "tangent alphabet"
as d!A! = {da_1, ... , da_n}, in principle, just an arbitrary alphabet of symbols,
disjoint from the initial alphaber !A! = {a_1, ..., a_n}, that is intended to be
interpreted in the way just indicated. It only remains to be understood that
the precise interpretation of the symbols in d!A! is often conceived to be
changeable from point to point of the underlying space A. (For all we know,
the state space A might well be the state space of a language interpreter,
one that is concerned, among other things, with the idiomatic meanings of
the dialect generated by !A! and d!A!.)
In the above terms, a typical "tangent space of A at a point x", frequently
denoted as T_x (A), can be characterized as having the generic construction
dA = <|d!A!|> = <|da_1, ..., da_n|>. Strictly speaking, the name "cotangent
space" is probably more correct for this construction, but the fact that we
take up spaces and their duals in pairs to form our universes of discourse
allows our language to be pliable here.
Proceeding as we did before with the base space A, we can analyze the
individual tangent space at a point of A as a product of distinct and
independent factors:
dA = Prod_i dA_i = dA_1 x ... x dA_n.
Here, dA_i is an alphabet of two symbols, dA_i = {(da_i), da_i},
where (da_i) is a symbol with the logical value of "not da_i".
Each component dA_i has the type B, under the correspondence
{(da_i), da_i} ~=~ {0, 1}. However, clarity is often served
by acknowledging this differential usage with a superficially
distinct type D, whose intension may be indicated as follows:
D = {(da_i), da_i} = {same, different} = {stay, change} = {stop, step}.
Viewed within a coordinate representation, spaces of type B^n and D^n may
appear to be identical sets of binary vectors, but taking a view at this
level of abstraction would be like ignoring the qualitative units and
the diverse dimensions that distinguish position and momentum, or
the different roles of quantity and impulse.
Jon Awbrey
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