Re: [ontolog-forum] Is there something I missed?

[(•`'·.¸(`'·.¸(•)¸.·'´)¸.·'´•) .,., <asaegyn@xxxxxxxxx>]

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> 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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