[Inquiry] Transitivity & Triadicity

Jon Awbrey jawbrey at att.net
Thu Feb 1 22:50:33 CST 2007


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I am bringing my previous notes on this subject together here
just so I'll be able to keep this series all in the same month.

Note 1

Just off-hand I would have to guess that transitivity is
an instance of degenerate thirdness, since the matter of
the triple x, y, z being in order reduces to the matters
of the pair x, y being in order and the pair y, z being
in order.

Note 2

Another example that comes to mind in the context
of value orderings is Warren S. McCulloch's paper,
"A Heterarchy of Values Determined by the Topology
of Nervous Nets", reprinted in ''Embodiments of Mind'',
MIT Press, 1965.  He discusses the order of complexity
in neural mechanisms that would be necessary to support
what he calls a "value anomaly", an intransitive or cyclic
triple where A is preferred to B, and B is preferred to C,
but C is preferred to A.

Note 3

A question has arisen about the relationship between
transitive relations and irreducibly triadic relations.

First off, we can easily make the following observations:

1.  A transitive relation is a dyadic relation.
2.  An irreducibly triadic relation is a triadic relation.

So it is safe to say that a transitive relation is never
to be classified as an irreducibly triadic relation.

Thus we must inquire deeper as to the nature
of the question that is being asked here.

Note 4

In preparing to write something on this theme, I discovered that I had
already written up an exposition that seems to bear some relation to it.
I've been converting this to wiki format at the GetMeta or GetWiki site:

http://www.getwiki.net/wiki.php?title=Propositional_Equation_Reasoning_Systems#Variations_on_a_theme_of_transitivity

It's still a bit rambling at present, but will hopefully become clearer over time.

Jon Awbrey

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