SUO: Re: General Design
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John F. Sowa wrote:
>
> Jon,
>
> That is certainly true:
>
JA: Now, I dearly love lattices, posets, preorders as much as
> > anybody has a right to, but they are all 2-adic relations,
> > and so there's just gonna have to be some threads hanging
> > loose when you try to pack up sign relations or any other
> > 3-adics in your lattices. It's the conservation of arity.
>
> Every lattice, tree, or other partial ordering is based
> on a dyadic relation. And every type label is defined
> by a monadic relation. And every proposition is a
> zero-adic relation (or medad in Peirce's terminology).
>
> So what?
Well, many of the "so whats" were supposed to be covered by the remarks you excised,
but I am nothing if not peircistent, so I will try to say it another way. Whenever
a situation is adequately modeled by a k-adic relation, for k = 0, 1, 2, then we are
well-advised to know that and to model it accordingly. But you are the one who keeps
on mentioning purpose, and adding a consideration of purpose to any previous situation
will up the arity by one. This actually comes up in applications that are far remote
from any acquaintance with Peirce. I remember one study that I became acquainted with
at the time when many behaviorists were shifting their paradigm to "animal cognition".
An animal that has to go from A to B in order to get to C is faced with the cognition
of a 3-adic relation, and when the 3-adicity is not degenerate -- what comes to mind
is when the ring wraiths had to spur their horses on to the next bridge in order to
get across the river -- then some critters will be competent to cognize it on their
own and some will not solve the problem without the prodding of an intelligent rider.
In many ways, this humble example epitomizes the whole Chomsky/Piaget/Skinner debate.
One way to think about the quantum leap between finite state and non-finite state
behavior is in the difference between stimulus-response matrices that represent
2-adic relations and, at the next step up, push-down stacks for context-free
grammars and languages, that involve an irreducible element of 3-adicity.
It is clear from the early papers of Arbib, Chomsky, McCulloch, Minsky,
Newell, Papert, Schutzenberger, Simon, and that whole crew that they
explicitly undertstood context-free parsing and up as modeling a type
of abductive, backtracking, guessing, hypothesis-forming-&-testing
level of complexity in competence and performance. So the upshot
is that accounting for purpose requires triple-entry bookkeeping.
This and a host of other examples force us to consider whatever order relations that
we were using before as "local resolutions" of the more generic 3-adic relations that
it actually takes to cover the complexity of the global situation. For instance, one
of the simplest ways of getting a local resolution is by fixing the purpose or the end
in view for the sake of a given discussion, but when you have occasion to think outside
that particular box, then you'll have to move back up to the full 3-adic relation again.
Jon Awbrey
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