Re: Interpretation of Relations

Joe Thurbon wrote:

> On 2007-01-23 12:23:37 +1000, "JOG" <jog_at_cs.nott.ac.uk> said:
>
> > Joe Thurbon wrote:
> >> [snip]
> >>> Say we have a proposition from
> >>> the real world, which has three roles x,y and z, and three
> >>> corresponding values a,b and c. RM as it stands would represent this
> >>> proposition directly as a tuple:
> >>>
> >>> P(x:a, y:b, z:c)
> >>>
> >>> whereas I believe a tuple should perhaps represent it 'indirectly' as
> >>> a compound predicate:
> >>>
> >>> Exists p x(p,a) ^ y(p,b) ^ z(p,b)
> >>>
> >>> I believe the consequences of this subtle change in interpretation of
> >>> what we are 'recording' (facts - or statements /about/ facts) _may_ be
> >>> able to remove a lot of the logical errors generated by missing
> >>> information, and perhaps some other issues too. But don't quote me on
> >>> that.
> >>
> >> I'm not sure exactly what the change in notation here is buying you.
> >> The formula still looks like facts, rather than statements about facts.
> >
> > Well comparing:
> > F1 = P(a, b, c)
> > F2 = Ep x(p,a) ^ y(p,b) ^ z(p,b)
> >
> > I'd certainly consider F1 and F2 different propositions.
>
> I was a bit terse (I was trying to get a reply in before work) - I
> didn't mean to suggest that the approaches were equivalent. In fact, I
> think there is some real merit in the F2 approach (if I understand it).
>
> One thing concerns me, though. Where do the 'p's come from? The only
> interpretation I can make of them are that they are the 'things' which
> the 'facts' are about. Or are you actually quantifying over predicates?
> (In which case you're using 2nd order logic).

Yes I'm intending to quantify over predicates, so absolutely 2nd order logic. This in turns gives me a lot of intuitive set theory notation if I can align it to make any sense.

>
> > F1 comments
> > about the real world directly, whereas in a sense F2 is commenting
> > about F1 (especially given it starts with existential quantifier,
> > 'there is a proposition').
>
> In FOL, you don't quantify over propositions, you quantify over
> constants. So you are talking about 2nd order logic (which is fine, I
> just know close to nothing about it). At this point, I'm actually
> unsure how to read F2 above.

I'd read F2 := Ep x(p,a) ^ y(p,b) ^ z(p,b) as "There is a proposition in which Role x has Value a, Role y has Value b and Role Y has value c".

>
> > It also incorporates attribute names
> > explicitly within the encoding (which seems to correspond to Codd's
> > move to 'relationships' between the 1969 and 1970 paper pretty well)
> > and it reflects the unordered nature of attributes in databases, given
> > the conjunctions are commutative.
>
> Yes, I like this.
>
> >
> > Either way, I think it would be more than a change in notation - the
> > database is no longer expected to comment reliably on the real world,
> > but rather just on what we know about the real world. If it stores one
> > fact "I have been given a proposition that states Joes hair is Red",
> > CWA over the database no longer implies that Joes hair is not black,
> > but rather just that "I have not been given a proposition that states
> > Joes hair is Black", saving me from any missing information
> > contradictions.
>
> I understand the intent. I think your translation is on the right path,
> but I think it's not quite what you intend.

If you could elaborate on this, then I'd appreciate that. I don't want to get caught up a blind alley logically.

>
> >
> > As a logician perhaps you can tell me if the following makes any sense
> > - to say Joe does have a hair colour and is not bald, I'd have:
> > P = Ep Name(p, Joe) ^ Ey Hair(p, y).
> >
> > Or to say Joe is bald:
> > P = Ep Name(p, Joe) ^ ~Ey Hair(p, y).
> >
> > Or to say I don't know if I don't know if Joe is bald or not, well this
> > is inferred from not saying anything at all.
>
> Yep, this makes sense to me. I'm assuming that by "=" you mean 'is
> defined as'. It's not really a part of FOL, that I'm aware of, but it
> makes sense.

Yes I'm using it as an alias. Perhaps := would be better notation.

>
> Just to clarify what I was talking about above, in this situation, Joe
> and p are not propositions, though, they are a constant and a variable.
> So, to relate this to the relation
>
> HasHair(Person:p, Hair:h)
>
> you have to do something more than just have an Ep around. HasHair is
> not in the domain of the Exists.

I need to mull on this.

>
> >
> > There are other possible consequences in terms of being able to more
> > advanced collectivizing of propositions but I'm nowhere on top of it
> > (having to pay a mortgage and all),
>
> I hear you!
>
> [...]
>
> Cheers,
> Joe
Received on Tue Jan 23 2007 - 15:38:09 CET