Re: Guessing?

From: Brian Selzer <brian_at_selzer-software.com>
Date: Mon, 2 Jun 2008 02:28:53 -0400
Message-ID: <FqM0k.2330$89.32@nlpi069.nbdc.sbc.com>

"Brian Selzer" <brian_at_selzer-software.com> wrote in message news:fNI0k.3402$xZ.760_at_nlpi070.nbdc.sbc.com...
>
> "David BL" <davidbl_at_iinet.net.au> wrote in message
> news:b7d01d30-8f85-406a-a07f-6801809a1a97_at_b5g2000pri.googlegroups.com...
>> On May 31, 5:11 pm, "Brian Selzer" <br..._at_selzer-software.com> wrote:
>>> "David BL" <davi..._at_iinet.net.au> wrote in message
>>>
>>> news: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?
>>>
>>> Sort of, but correct me if I'm wrong: the intension of a relation
>>> shouldn't,
>>> and barring schema evolution, doesn't, change with time, whereas the
>>> extension usually can and does as what is to be represented in the
>>> relation
>>> comes into being, changes in appearance or ceases to exist.
>>
>> An intensional definition can be a function of time. Eg
>>
>> S(t) = set of surnames of UK prime-ministers
>> after Thatcher at time t
>>
>> with (current) extension
>>
>> S(June 2 2008) = { Major, Blair, Brown }
>>
>> Knowing the extension at a particular time doesn't tell you what the
>> extension is at other times.
>>
>> I guess you could say the intensional definition doesn't change with
>> time because t is bound! Is that what you mean?
>>
>>
>
> No. The intension states what can be. It cannot by itself state what is
> unless what is represented is necessarily the case.
>

Just to clarify: I don't think the intension can be a function of time because it cannot be known what /will/ happen in the future. It can, however, be known what /can/ happen in the future. The intension specifies what is possible. This specification is not a function of time: it is independent of time. The set of all possible database values does not change with time. Which of those possibilities reflects what is actually the case is what does. Of course if there is only one possibility, which can only be the case if what is the case is /necessarily/ the case, then and only then does the intension by itself state what is.

>>> The
>>> determination that something /actually/ exists exceeds what the
>>> intension by
>>> itself can provide but not what the extension due to Domain Closure can
>>> provide; on the other hand, the intension does specify what can be
>>> represented and indirectly, therefore, again due to Domain Closure
>>> whether
>>> something /can/ exist. To determine from the intension whether
>>> something
>>> /actually/ exists would require an interpetation. So yes, the intension
>>> relates back to the real world. It should be noted here that the
>>> extension
>>> also relates back to the real world but that Domain Closure makes it
>>> possible to draw conclusions from the data without resorting to
>>> interpretation. I would argue, therefore, that the intensional
>>> definition
>>> /is/ part of the RM formalism, and that the mechanism of interpretation
>>> is
>>> also part of the RM formalism, but that any particular interpretation is
>>> not. The predicate of a database, including all state constraints,
>>> determines the set of all possible database values, but it is only under
>>> an
>>> interpretation that one of those possibilities can be designated to be
>>> actual.
>>
>> Sorry I don't understand this. For example what does "actually
>> exist" mean?
>>
>
> Though there may be many possible database values, only one can represent
> what is actually the case. And only that which is represented in that
> database value actually exists. If you're curious, look up Domain
> Closure.
>
>
Received on Mon Jun 02 2008 - 01:28:53 CDT

Original text of this message