Re: SUO: RE: Re: Focus and Volume
>Phil, Jon, Jim, et al.,
>
>I agree that the points Jon A. is trying to make are often obscure,
>and I also agree that Jon would make his case much stronger if
>he would state what to him may seem obvious. But I also believe
>that the Donald Duck example is very pertinent to the disucssion.
>
>Philip Jackson wrote:
>
> > Perhaps Jon can explain the relevance of his postings which seem quite
> > frivolous, such as information about Donald Duck's nephews...
Really, shouldnt the SUO list provide enough evidence that it is
unwise to suggest that Jon explain *anything*?
>Following is the relevant excerpt, stated in KIF:
>
>1.
>
>| (forall (?x ?y ?z)
>| (<=> (TitanicTrio ?x ?y ?z)
>| (exists (?e)
>| (and (TitanicTrio ?e)
>| (Member1 ?e ?x)
>| (Member2 ?e ?y)
>| (Member3 ?e ?z)
>| ) ) ) )
>
>2.
>
>| (forall (?x ?y ?z)
>| (<=> (TitanicTrio ?x ?y ?z)
>| (exists (?e)
>| (and (Donald ?e)
>| (1st Nephew ?e ?x)
>| (2nd Nephew ?e ?y)
>| (3rd Nephew ?e ?z)
>| ) ) ) )
>
>These two KIF statements have exactly the same structure, except that
>the first one assumes the existence of an abstract entity ?e of type
>TitanicTrio (i.e., something very much like a set, which most of us
>have agreed to be abstract) while the second one assumes the existence
>of an entity ?e that happens to be a duck named Donald.
>
>I believe that this example is a good answer to the claim that
>triadic relations can be reduced to dyadic relations by postulating
>the existence of some new entity ?e. Although that point may be
>formally true, Jon's example shows that there is no way of knowing
>whether that entity is something abstract, such as a set or some
>hypothetical TitanicTrio, or something concrete such as a Duck.
I had not seen this before (all email from JA is filtered out of my
SUO mailbox), but it deserves some comment. In fact, the claim that
is risky here is that the first of these refers to an abstract entity
and the second to a duck. That might be what the writer intended, but
it isn't what the KIF says. Being isomorphic, these two pieces of
KIF, taken in isolation, have precisely the same models: any
interpretation that makes one true can be easily transformed into an
isomomorphic interpretation, over the same universe, that makes the
other true.
I presume that what John means is that the *intended*
intepreretations are different in the way he describes. But the
intentions of the writers of axioms have no place in any kind of
formal analysis of the meanings of those axioms (other than by
exclusion, in the sense that one can formally show that the actual
meaning does not correspond to the intended meaning, cf. the previous
paragraph.) So this kind of example can hardly be claimed by Jon (or
anyone else) to be any kind of *demonstration* of some kind of
irreducibility. At most, it could show that some distinction that Jon
(or someone) has in mind, cannot be represented in the formal
notation. To get from that to an irreducibility result requires
showing that the distinction *can* be represented in some other
formalism. Good luck with the mathematical theory of ducks, whoever
sets out to show this.
Pat Hayes
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