From the point of view of having something that can be put in front of
domain experts for review, I would rather have a ternary relation than
the mathematical solution to the problem of reducing it to binary
relations. It's one thing, as John says, that OWL falls down on. (01)
Mike (02)
•`'·.¸(`'·.¸(•)¸.·'´)¸.·'´• wrote:
> John and Pat,
>
> While I realize that it is possible to reduce many higher valence
> expressions to those with lower arity, it is unclear to me why we
> would force this on people.
>
> As John illustrated below, it /is /possible to capture "+" as a series
> of binary and unary relations, though the syntax and articulation of
> this idea seems rather unnatural and unintuitive.
>
> Off the top of my head, my location seems like a quaternary relation
> (location Ali, x, ,y ,z) in 3D space. Or by GPS, it'd be at least a
> ternary relation. Similarly, if i wanted a time stamp associated
> /within/ a relation (as opposed to a conjunction with another
> relation), i'd want potentially higher arity relations.
>
> To use an analogy, while i might be able to reconstruct a
> multi-variable derivative by taking partial derivatives and then
> taking a series, it seems like an awkward workaround to express what
> would otherwise be a straightforward oncept.
>
> Given that this discussion is ostensibly concerned with
>
> Is there something I missed? (and What is an Ontology)
>
> Might you tell me what the advantage of restricting vocabularies to
> unary and binary predicates are?
>
> This seems to be steering the discussion to the question of "what
> constitutes a '/good/' axiom?"
>
> //
>
> Indeed, it seems to me that the greatest difficulty in the creation of
> ontologies is the paucity of guidelines as to what constitutes a
> "good" axiom or ontology. Michael Uschold and Michael Gruninger wrote
> a paper briefly touching this topic in 1996, though i'm not sure how
> much of an impact it had (277 citations). They appealed to the notion
> of /competency questions/ to guage whether the ontology you have
> developed is addressing its purported function.
>
> Extending this idea, if an ontology is a coherent account of what
> (relevantly) is, in some formal language, we should be concerned with
> capturing that knowledge in a direct way. If we so desire, we may then
> use projection to reduce the arity of the relation (and perhaps create
> contexts), but to /a priori/ restrict expression and understanding of
> ontologies to this particular mode of representation seem odd to me,
> unless of course, i'm missing something :P.
>
> Ali
>
> --
> Ref - Uschold & Gruninger 1996) M. Uschold and M. Gruninger.
> "Ontologies: Principles, methods and applications." Knowledge
> Engineering Review vol. 11, pages 93-196, 1996.
>
> On Thu, Jan 29, 2009 at 12:48 PM, John F. Sowa <sowa@xxxxxxxxxxx
> <mailto:sowa@xxxxxxxxxxx>> wrote:
>
> Ali and Pat,
>
> I agree with Pat's comments on this topic, but I'd like to
> add a few.
>
> Nicola G> This set of assumptions has usually the form of a
> > first-order logical theory, where vocabulary words appear as
> > unary or binary predicate names, respectively called concepts
> > and relations.
>
> AH> Though i'm not sure why vocabulary words are restricted to
> > unary or binary predicate names
>
> PH> Me neither. Writing in 2008, Nicola was probably intending to
> > make a nod at the prevailing widespread use of description
> > logics, which are restricted to the unary/binary case.
>
> As an example, the English word 'add' maps to the mathematical '+',
> which represents a function with two inputs and one output.
>
> It is possible to represent such things with a concept type Add
> that represents an add operation, which is linked to three dyadic
> relations:
>
> 1. Arg1 links the Add concept to the first argument.
>
> 2. Arg2 links Add to the second argument.
>
> 3. Rslt links Add to the result.
>
> PH> In practice, there seems to far more unary/binary than anything
> > else, and one can routinely encode an n-ary relation is a
> > conjunction of binary ones.
>
> Yes. And it's desirable to use a logic that can relate the different
> representations by if-then rules, such as
>
> (forall (x y z w)
> (if (and (Add w) (Arg1 w x) (Arg2 w y) (Rslt w z))
> (= z (Sum x y)) ))
>
> PH> Most ontologies are built on a skeleton of taxonomy, or at least
> > a subclass hierarchy. Its hard to avoid having such a structure
> > somewhere in any large ontology, in fact.
>
> Yes. That has been common practice since Aristotle. In fact, many
> of the ontologies that are written in OWL don't use anything beyond
> Aristotle's subset.
>
> John Sowa
>
>
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