[Inquiry] Re: Futures Of Logical Graphs

[Jon Awbrey]

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FOLG.  Note 11

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Cybernetics List, Peirce List,

As agents of systems, whether that system is our own physiology
or our own society, we move through what we commonly imagine to
be a continuous manifold of states, but with distinctions being
drawn in that space that are every bit as compelling to us, and
often quite literally, as the difference between life and death.
So the relation of discretion to continuity is not one of those
issues that we can take lightly, or simply dissolve by choosing
a side and ignoring the other, as we may imagine in abstraction.
I'll try to get back to this point later, one in a long list of
cautionary notes that experience tells me has to be attached to
every tale of our pilgrimage, but for now we must get under way.

Returning to En and Ex, the two most popular interpretations
of logical graphs, that happen to be dual to each other in a
certain sense, let's see how they fly as "hermeneutic arrows"
from the syntactical domain !S! to the objective domain !O!,
at any rate, as their trajectories can be spied in the radar
of what George Spencer Brown called the "primary arithmetic".

Taking En and Ex as arrows of the form En, Ex : !S! -> !O!,
at the level of arithmetic taking !S! = {rooted trees} and
!O! = {Falsity, Truth}, let's factor each arrow across the
domain of formal constants !S!_0 = {O, |}, the domain that
consists of a single rooted node plus a single rooted edge.
As a strategic tactic, this allows each arrow to be broken
into a purely syntactic part En_syn, Ex_syn : !S! -> !S!_0
and its purely semantic part En_sem, Ex_sem : !S!_0 -> !O!.

As things work out, the syntactic factors are formally the same,
leaving our dualing interpretations to differ in their semantic
components alone.  Specifically, we have the following mappings:

   En_sem : O ~> false, | ~> true

   Ex_sem : O ~> true,  | ~> false

On the other side of the ledger, because the syntactic factors,
En_syn and Ex_syn, are indiscernible from each other, there is
a syntactic contribution to the overall interpretation process
that can most readily be investigated on purely formal grounds.
That will be the task to face when next we meet on these lists.

Jon Awbrey

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