SUO: Irreducible Triads
Rich and Jon,
This issue has been hashed, rehashed, and made
into mincemeat time and again.
JA> The term "thirdness" refers to those properties of both
> empirical phenomena and formal structures that can only
> be modeled in any adequate way by means of 3-adic relations.
RC> There are no such formal structures, IMHO. John McCarthy's
> paper "Recursive Functions of Symbolic Expressions", which
> provides the mathematical concept behind the first Lisp
> interpreters, indicates that any function, of any arity,
> can be modeled by a decomposition of the same mapping
> into dyadic and monadic functions.
The attached .GIF file illustrates the conversion that
John M. and many others (including me) have performed
many times over.
Notice how it eliminates the triadic relation of type
Gives by replacing it with a concept node of type Give
and three dyadic relations of type Agnt, Thme, and Rcpt.
It merely replaces one kind of triad with a different
kind of triad. The underlying graph structure still
has an irreducible triad.
Peirce, McCarthy, and many other mathematicians and
logicians who have commented on this point have all been
correct. But they have all been talking past one another.
Peirce said that you cannot remove an irreducible triadic
connection from a graph, and McCarthy and others have said
that you can replace a triad of one kind with a triad of
another kind.
Bottom line: Thirdness is Thirdness. You can push it
around all you like, but when you think you've squeezed
it out from one spot, it pops up somewhere else.
John Sowa
