# [sc34wg3] RM4TM and symmetrical relations

**Bernard Vatant
**
sc34wg3@isotopicmaps.org

*Wed, 20 Nov 2002 15:59:56 +0100*

[Bernard Vatant]
>* > how do I express sibling-ness or any other equivalence relationship if:
*>* > 1. A single membership is not allowed.
*>* > 2. Having two c-nodes of the same type in an assertion is not alllowed.
*
[Steve Newcomb]
>* By "equivalence relationship" I'm going to assume that
*>* you mean "relationships in which there's no semantic
*>* difference between the role types". Such as opposites.
*
Not exactly. "equivalence relationship" or more exactly "equivalence relation" has in my
mind the meaning it has in mathematics, a relation R which has the three properties of
reflexivity, symmetry and transitivity.
Which means, for all x, y, z in the domain of the relation:
1. xRx (reflexivity)
2. If xRy, then yRx (symmetry)
3. If xRy and yRz, then xRz (transitivity)
In that sense in fact sibling-ness is an equivalence relation only if it's understood as
"belong to the same set-of-siblings" (I do not find any substantive in english for the
french "fratrie", meaning the set of siblings in the same family.)
In that extended definition, I am a sibling of myself, though this can be questionable of
course.
But the exemple you give ("opposites") is definitely not an equivalence relation, it lacks
reflexivity and transitivity.
A true equivalence relation in geometry is for example "is parallel to" (for straight
lines) more exactly expressed as "has the same direction as". If, to express the assertion
"D1 is parallel to D2", I need two different role types, I give up, and your long
explanation keeps me only wondering.
If indeed the Subject Location Uniqueness Principle leads to such convoluted and
non-intuitive conclusions ...
... either you got wrong in your reasoning (and I can't really judge on that, I'm afraid,
because I'm lost in your arguments)
... or the principles that lead to such conclusions are questionable.
Bernard