ONT Re: Inquiry Into Irreducibility
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Howard,
You now have a stock of concrete examples and diverting discussions
for illustrating various issues in the ir/reducibility of relations.
Let me now try to give a summary account of the high points, and also
the sundry pitfalls, that I have learned will typically be encountered
as one attempts to grapple with these issues. First, let me apologize
in advance for even mentioning a number of these potential confusions,
as not a few of them will appear too obvious to ever trap any wary
explorer, but I promise you, I would not be bringing them up if
I had not seen their impedimenta actualized on many occasions.
1. It is very important to distinguish relations,
as objects of discussion and thought, from the
signs of relations, that is, for the most part,
the names, expressions, concepts, and formulas
that we use to talk and to think about them.
2. It is very important to distinguish a relation,
as a whole entity, from any one of its elements,
that is, the tuples that fall under the relation.
The importance of this is independent of whether
one works with the extensions or the intensions
of the relations in question.
3. When Peirce spoke of relations being reducible or not
he was speaking of reducibility under composition, or
what he called "relative multiplication". This may be
viewed as the logical analogue of matrix multiplication,
with the logical operations of conjunction and disjunction
replacing the algebraic field operations of multiplication
and addition, respectively. From a logical point of view,
this is the principal notion of reduction among relations.
With regard to the definition of relational composition,
it is as mistaken to think that one can multiply 2-adic
relations and obtain a 3-adic relation as it would be
to think that one can multiply square matrices and
come up with a cubic array. On the other hand,
arity 3 is the breakpoint: Any k-adic relation,
for k > 3, can be decomposed into relations of
arity 3 or less.
4. There is also the notion of "projective reducibility",
or "reconstructibility of relations from projections",
that can be employed to speak of some 3-adic relations
being "reducible" or "reconstructible" in this sense,
and others not, so long as one understands that a few
special 3-adic relations, like those intrinsic to the
use of logical connectives, are being used in the very
process of achieving the reductions and reconstructions.
Jon Awbrey
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