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Re: [ontolog-forum] borrowing terminology

To: <conrad.bock@xxxxxxxx>
Cc: "'[ontolog-forum] '" <ontolog-forum@xxxxxxxxxxxxxxxx>
From: Pat Hayes <phayes@xxxxxxx>
Date: Fri, 9 Mar 2007 15:59:43 -0600
Message-id: <p06230915c217805d3ea9@[]>
>  >  >Not following, the definitions in the math links say all elements
>  >  >in the range have mappings from some element in the domain, while
>  >  >the co-domain can have elements that aren't mapping to by any
>  >  >element in the domain.
>  >  Yes. But at least the issue is about sets and
>  >  relations. The definitions for 'range' in the
>  >  wikipedia CS entry (pointer above) are to do with
>  >  boundary values in arrays and limits on values of
>  >  variables, which are a COMPLETELY different
>  >  topic. I thought this was your point.
>I take variables as properties (in the OWL sense) of processes as they
>are executing (like the processes you see in the task list of operating
>systems).    (01)

Wow, I hadn't thought of that particular 
interpretation. OWL properties are inherently 
'static', though.    (02)

>  Their types are the ranges (in the OWL sense) of those
>properties.    (03)

I see the point (now :)    (04)

>  >  >In OWL, if the range of a property pet is Animal, it isn't
>  > required that all animals are pets.
>  >  Right, so the owl:range is a subset of the co-domain, as I said.
>I think it;s the (Wiki) co-domain:
>   http://en.wikipedia.org/wiki/Codomain
>   (Wolfram doesn't give a term for this concept, as far as I can tell).    (05)

Well, the OWL (actually RDFS, which OWL also 
uses) range isn't exactly either the math-range 
or the math-co-domain. Both of these are pinned 
down exactly, whereas the RDFS/OWL notion is 
sloppier, it represents a constraint on the 
"real" range rather than specifying it exactly. 
Similarly for the RDFS/OWL notion of domain, by 
the way. (This was done because it makes life 
easier for users and it keeps the logic 
monotonic, neither of which matter a damn in 
mathematics.)    (06)

>The co-domain is the second element in the cross product that defines
>the tuples of the relation.  The domain is the first.    (07)

Right. OWL uses "range" and "domain" 
respectively, which is an older mathematical 
convention. The word "co-domain" really only 
makes sense within category theory, speaking 
strictly.    (08)

>  The
>(Wiki/Wolfram) range is are all and only those elements in the co-domain
>that are mapped to from some element in the domain.    (09)

Right. But this wouldn't make sense in OWL since 
there is no way to 'pin down' the exact domain 
either, for the same reason.    (010)

>In the above example, the pet property has co-domain Animal, and its
>range is the Pet class (qua-type in KL-one).
>  >  In fact, if S is any set which is a subset of the codomain and which
>  >  has the math-range as a subset, then it would be correct to assert
>  >  that S is an owl-range for the property.
>Sure, you can always limit the co-domain to the range without ill
>effect, but the typically the co-domain is used as a way of specifying
>what individuals *can* be values of the property, which the range
>doesn't tell you.    (011)

Well, If that is what it means, then I for one am 
glad to get rid of it :-) The last thing we need 
in ontology languages are modalities.    (012)

>I've heard of approaches that always use the range as
>the co-domain, and assume "reclassification" of an individual as a
>member of the range class is restricted by some other means.
>  >  >Not the way I've heard idempotence used by software people.
>  >  >They're referring to calling the same function twice on the same
>  >  >arguments (the wikipedia ambiguously calls this "used multiple
>  >  >times"), where the second call has no effect.
>  >  Ah, I see your point, sorry. Hmm.  I think I see where this
>  >  happened.  If you assume that every application involves a 'state',
>  >  then multiple applications become nested function applications to a
>  >  starting state, which maps the computer science sense into the
>  >  mathematical sense.
>And even then you need to assume the CS functions return the system
>state, which of course, they don't.    (013)

True, but LISP and continuation semantics cast a very long shadow :-)    (014)

>  >  >I thought it equivalent to a set of disjoint classes that are also
>  >  a union for another.  What's the mathematical definition?
>  >  It unions distinct 'copies' of the sets, so even if they are
>  >  identical or share elements, one gets distinct copies in the
>  >  disjoint union (aka 'separated union'). This can be described
>  >  formally in various ways, but the key property is that the
>  >  cardinality of the disjoint union is always the sum of the
>  >  cardinalities of the unioned sets. So the n-fold disjoint union of A
>  >  with itself is meaningful and isomorphic to A x n.  If the sets are
>  >  disjoint, then the disjoint union is the ordinary union. But it is
>  >  defined even if they aren't disjoint; and this is important for the
>  >  category-theory treatment of set theory.
>I see, the category people are assuming bags.    (015)

Well, not exactly, though that is a nice CS way 
to put it in a nutshell. Done strictly, it 
applies to an indexed family of sets, so applying 
it to simple sets is a "familiar abuse of 
notation" in mathspeak.    (016)

>  But union can apply to
>sets, and the individuals conforming to an OWL class are a set, aren't
>they?    (017)

Yes. There is no doubt what OWL needs: but they 
shouldn't *call* it "disjoint union". What it is, 
is regular union, but in addition assuming the 
unionees are disjoint: that is, it makes an 
implicit assertion (which is at best risky in 
such a delicately tuned assertional language as 
OWL-DL) and is only a partially defined as an 
operation (which is generally a bad idea in a 
Boolean logic).    (018)

>BTW, I remembered a case of being overly careful in borrowing
>terminology.  The UMLers decided to use the term Multiplcity instead of
>Cardinality for the restrictions on the number of values of a property,
>on the grounds that "cardinality" means the number of values in set,
>whereas what is needed is a constraint on those.  This causes
>unfortunate disconnects when UMLers talk with database modelers and
>OWLers.    (019)

Ah, sigh. My favorite example of misappropriation 
of terminology is the inhumanly ugly XML use of 
the word "entity", which has now permeated all of 
the W3C internal discourse. But a close second is 
the inflation of the meaning of "resource" from 
its original use by the Internet pioneers, where 
it was close to the normal meaning, to mean 
"anything whatsoever", i.e. 'entity', in 
W3C-speak.    (020)

Pat    (021)

>Conrad    (022)

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