[Inquiry] Re: Differential Logic

[Jon Awbrey]

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DLOG.  Note D6

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Basis Relativity and Type Ambiguity

Finally, two things are important to keep in mind with regard to the
simplicity, linearity, positivity, and singularity of propositions.

First, all of these properties are relative to a particular basis.
For example, a singular proposition with respect to a basis !A!
will not remain singular if A is extended by a number of new
and independent features.  Even if we stick to the original
set of pairwise options {a_i} |_| {(a_i)} to select a new
basis, the sets of linear and positive propositions are
determined by the choice of simple propositions, and
this determination is tantamount to the conventional
choice of a cell as origin.

Second, the singular propositions B ::> B, picking out as they do a single cell
or a coordinate tuple of B^n, become the carriers or the vehicles of a certain
type-ambiguity that vacillates between the dual forms B^n and (B^n ::> B) and
infects the whole hierarchy of types built on them.  In plainer language, the
terms that denote the interpretations x : B^n and the singular propositions
x : Bn ::> B are fully equivalent in information, and this means that every
token of the type B^n can be reinterpreted as an appearance of the subtype
B^n ::> B.  And vice versa, the two types can be exchanged with each other
everywhere that they turn up.  In practical terms, this allows the use of
singular propositions as a way of denoting points, forming an alternative
to coordinate tuples.  For example, relative to the universe of discourse
[a_1, a_2, a_3] the singular proposition a_1 a_2 a_3 : B^3 ::> B could be
explicitly retyped as a_1 a_2 a_3 : B^3 to indicate the point ‹1, 1, 1›,
but in most cases the proper interpretation could be gathered from context.
Both notations remain dependent on a particular basis, but the code that is
generated under the singular option has the advantage in its self-commenting
features, in other words, it constantly reminds us of its basis in the process
of denoting points.  When the time comes to put a multiplicity of different bases
into play, and to search for objects and properties that remain invariant under the
transformations between them, this infinitesimal potential advantage may well evolve
into an overwhelming practical necessity.

Jon Awbrey

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