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## Re: [ontolog-forum] Question: more on N-ary?

 To: "[ontolog-forum] " Chris Menzel Thu, 14 Jul 2005 18:43:09 -0500 <20050714234309.GV903@xxxxxxxx>
 ```On Thu, Jul 14, 2005 at 03:42:00PM -0700, Duane Nickull wrote: > Can anyone provide a clear concise distinction between a symmetric and > coreflexive relationships? The gist seems to be that both express a > relationship wherefor all x and y in X it holds that if xRy then x = y. > It appears that one states "holds xRy and yRx" while the other equates > equality "x=y"    (01) Symmetry and coreflexivity are properties of binary relations. A binary relation R is symmetric just in case, whenever it holds of the pair (a,b) it also holds of (b,a); formally:    (02) (<=> (Symmetric R) (forall (?x ?y) (=> (R a b) (R b a))))    (03) Note this is conditional -- a symmetric relation R needn't hold between all, or even any, pairs (a,b); but whenever it *does* hold of (a,b), it must hold of (b,a) as well. An example of a symmetric relation is "brother-of".    (04) A relation is coreflexive just in case it holds of (a,b) only if a is identical to b:    (05) (<=> (Coreflexive R) (forall (?x ?y) (=> (R a b) (= a b))))    (06) Coreflexive relations are pretty rare in formal ontology, as they are, as Duane notes, simply a restricted form of identity, and there aren't many (any?) intuitive relations that play that role. It's easy enough to cook one up, of course:    (07) (<=> (CabbageIdentical ?x ?y) (and (Cabbage ?x) (Cabbage ?y) (= ?x ?y)))    (08) Thus, a and b are CabbageIdentical just in case they are the same head of cabbage, and that is obviously coreflexive. But it's hard to see how such a relation could be useful for anything. The broader property of reflexivity is much more common and much more useful:    (09) (<=> (Reflexive R) (forall (?x) (R ?x ?x)))    (010) > Also, in SUMO, InverseRelation is expressed as: > > (=> > (inverse ?REL1 ?REL2) > (forall (?INST1 ?INST2) > (<=> > (holds ?REL1 ?INST1 ?INST2) > (holds ?REL2 ?INST2 ?INST1)))) > > The distinctions between the 3 seem to be very subtle. InverseRelation > and AsymmetricRelation are opposite? I find it a bit confusing.    (011) The inverse R' of a relation R holds of the pair (b,a) if and only if R holds of (a,b). Thus, if (as in set theory) we simply identify a binary relation with the set of ordered pairs it holds of, the inverse of a relation R is the relation that results from "flipping" each pair in R so that the first element of the pair becomes the second and vice versa.    (012) Unlike symmetry and coreflexivity, which, as noted, are *properties* of binary relations, inverse is itself a *binary relation* between two binary relations.    (013) Chris Menzel    (014) _________________________________________________________________ Message Archives: http://ontolog.cim3.net/forum/ontolog-forum/ Subscribe/Unsubscribe/Config: http://ontolog.cim3.net/mailman/listinfo/ontolog-forum/ Shared Files: http://ontolog.cim3.net/file/ Community Wiki: http://ontolog.cim3.net/wiki/ To Post: mailto:ontolog-forum@xxxxxxxxxxxxxxxx    (015) ```
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