SUO: Re: Irreducible Triads
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Brief Note, Brief Life.
McCarthy is talking about functions, not relations in general,
and he is invoking the idea that is called "Currying" functions
(after Curry's lambda calculus, but going back to Schoenfinkel's
combinators), namely, the fact that a function f of k arguments
can be represented in terms of a function f* from 1 argument to
a function of (k-1) arguments. No arity has been reduced in the
making of this curry. And it does not even touch the question
of the more general composition of k-adic relations.
see:
| Hindley & Seldin, 'Introduction to Combinators and Lambda Calculus',
| London Math Society Student Texts 1, Cambridge University Press,
| Cambridge, UK, 1986.
The irreducibility of 3-adic relations was established by Peirce in two senses,
which may conveniently be called "compositional" and "projective" reducibility.
All 3-adic relations are irreducible in the first sense, by the very
definition of composition. This is a simple fact, quite analogous
to the definition of the ordinary matrix product, inasmuch as any
2-adic relation can be represented as a rectangular matrix with
boolean coefficients, under the appropriate product rule, and
multiplying two such matrices always produces another, and
never a 3-dimensional array. By definition.
Some 3-adic relations are reducible in the second sense,
but many of the most important ones are not. These are,
again, simple mathematical facts, subject to elementary
proof, frequently by way of easy-to-construct examples.
From what I have been able to tell, people who fail to find all of
these simple mathematical facts immediately, well, maybe eventually,
obvious, are probably prone to at least one of three misconceptions:
(1) confusing a relation with a syntactic representation of it,
(2) confusing a relation with one of its tuples,
(3) drawing a factitious distinction between
logical connectives and logical predicates,
which appears to derive in part from (1).
So watch out for that.
John F. Sowa wrote:
>
> 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.
Unfortunately, the cogency of the graphical syntax appears to be understood only by
those who already understand what a k-adic relation is, namely, a set of k-tuples.
The graph is not the relation, though, it is only a syntactic representation.
> 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
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