ONT Identity & Teridentity

[Jon Awbrey <jawbrey@oakland.edu>]

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| Relations in the sense here considered are known, more particularly,
| as 'dyadic' relations;  they relate elements in pairs.  The relation of
| giving (y gives z to w) or betweenness (y is between z and w), on the other
| hand, is triadic;  and the relation of paying (x pays y to z for w) is tetradic.
| But the theory of dyadic relations provides a convenient basis for the treatment
| also of such polyadic cases.  A triadic relation among elements y, z, and w might
| be conceived as a dyadic relation borne by y to z;w [the ordered pair (z, w)].
|
| Quine, 'Math Logic', p. 201
|
| W.V. Quine,
|'Mathematical Logic, Revised Edition,
| Harvard University Press, Cambridge, MA, 1981.

This is one of the 'loci' that is frequently cited in claiming
that triadic relations decompose or reduce to dyadic relations.

What Quine fails to articulate here is that the general relation
of ordered pairs to their components is itself a triadic relation.

In particular, it has this shape:

o---------o---------o---------o
| First   | Second  | Pair    |
o---------o---------o---------o
| a       | a       | (a, a)  |
| a       | b       | (a, b)  |
| a       | c       | (a, c)  |
| ...     | ...     | ...     |
| b       | a       | {b, a)  |
| b       | b       | (b, b)  |
| b       | c       | (b, c)  |
| ...     | ...     | ...     |
o---------o---------o---------o

In graphical form we can draw each pairing-triple like this:

 x   y
  \ /
   o 
   |
 (x,y)

Thus, the ordered triple (u, v, w) of simple elements u, v, w
can be represented in the following two-step fashion:

 u   v
  \ /
   o 
   |
 (u,v)  z
   \   /
    \ /
     o
     |
     |
 ((u,v),z)

The moral of the story is that there is no way
to achieve synthesis without nodes of degree 3.

This has turned out to be a commonplace but a very fundamental fact
in many areas of math and computer science, theoretical and applied.

Jon Awbrey

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