⚠ It's A Trap ⚠

Post   : ⚠ It's A Trap ⚠
URL    : http://inquiryintoinquiry.com/2013/05/18/%e2%9a%a0-its-a-trap-%e2%9a%a0/
Posted : May 18, 2013 at 11:00 pm
Author : Jon Awbrey

Re: Kenneth W. Regan
At: http://rjlipton.wordpress.com/2013/05/15/graduate-student-traps/

The most common trap I run across has to do with Triadic Relation Irreducibility,
as noted and treated by the polymath C.S. Peirce.

This trap lies in the mistaken belief that every 3-place (triadic or ternary) relation
can be analyzed purely in terms of 2-place (dyadic or binary) relations — “purely” here
meaning without resorting to any 3-place relations in the process.

A notable thinker who not only fell but led many others into this trap is none other than
René Descartes, whose problematic maxim I noted in the following post.

• Château Descartes (http://inquiryintoinquiry.com/2013/02/14/chateau-descartes/)

As mathematical traps go, this one is hydra-headed.

I don’t know if it’s possible to put a prior restraint on the varieties of relational reduction
that might be considered, but usually we are talking about either one of two types of reducibility.

Compositional Reducibility.  All triadic relations are irreducible under relational composition, since
the composition of two dyadic relations is a dyadic relation, by the definition of relational composition.

Projective Reducibility.  Consider the projections of a triadic relation L \subseteq X \times Y \times Z
on the coordinate planes X \times Y, X \times Z, Y \times Z and ask whether these dyadic relations uniquely
determine L.  If so, we say L is projectively reducible, otherwise it is projectively irreducible.

See the MyWikiBiz or PlanetMath entries on Relation Reduction for further discussion —
• http://mywikibiz.com/Relation_reduction
• http://planetmath.org/RelationReduction

Previous Posts on Triadic Relation Irreducibility —
• http://inquiryintoinquiry.com/2013/01/17/triadic-relation-irreducibility-1/
• http://inquiryintoinquiry.com/2013/01/18/triadic-relation-irreducibility-2/
• http://inquiryintoinquiry.com/2013/05/16/triadic-relation-irreducibility-3/

-- 

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