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)
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