Re: Guessing?

From: David BL <davidbl_at_iinet.net.au>
Date: Fri, 30 May 2008 18:41:17 -0700 (PDT)
Message-ID: <5dc1c57d-8508-4180-a26c-38f944cf2779_at_q24g2000prf.googlegroups.com>


On May 30, 9:14 pm, "Brian Selzer" <br..._at_selzer-software.com> wrote:
> "David BL" <davi..._at_iinet.net.au> wrote in message

> > It seems to me that every base relvar will in practice have some
> > defined intensional definition outside the RM formalism and
> > inaccessible to the DBMS.
>
> I thought the intension of a relation states what can be while the extension
> states what is: wouldn't that place the intensional definition inside the RM
> formalism? I understand what you're driving at, though, but I think it is
> indeed a part of the RM formalism. Let me explain. Suppose you have
> predicate symbols P and Q. Isn't it true that under a first order logic
> interpretation, not only constant symbols are assigned meaning, but also
> predicate symbols? Isn't one of the assumptions under which the Relational
> Model operates the Unique Name Assumption? Wouldn't that assumption apply
> with equal force to predicate symbols as it does to constant symbols? What
> I mean by that is that it should not be possible for two predicate symbols
> to be assigned exactly the same meaning in the same way that it should not
> be possible for two constant symbols to be assigned exactly the same
> meaning. Now, a predicate can be a conjunction of other predicates, and the
> components of that conjunction can appear in other predicates, but if two
> predicates are composed of the exact same components, then they are really
> just one, and the Unique Name Assumption would require that only one
> predicate symbol be used to represent that particular conjunction of
> components. Bottom line: the name assigned to a relation is significant
> because it is a symbol for a distinct predicate.

Very informally I think an intensional definition of a finite relation should be sufficient to allow an omniscient being to calculate the corresponding extension. Therefore both intensional and extensional definitions state "what is". The difference comes down to whether the elements are explicitly enumerated.

In some cases an intensional definition of a set can be mathematically precise. Eg

        X = { x in Z | x < 100 and exists y in Z st x = y^2 }

which has an equivalent extensional definition

        X = { 0,1,4,9,16,25,36,49,64,81 }

However such mathematically defined sets aren't of primary interest to the RM - because the RM is mostly interested in recording finite sets that cannot be algorithmically compressed. Therefore extensional definitions are more important than intensional definitions. [On a side note - it seems to make sense to allow mathematically defined relations as read only first class citizens so that selection is just a join].

In practise the base relvars have intensional definitions that relate back to the real world and are outside our mathematical formalisms. Therefore a formal definition of the set is necessarily extensional.

That being said I generally agree with your above comments except I think it is more accurate to say that the assumption that base relvars have distinct associated intensional definitions is part of the RM formalism whereas the intensional definitions themselves are not. Does that make sense? Received on Sat May 31 2008 - 03:41:17 CEST

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