On Jan 21, 2011, at 9:46 AM, doug foxvog wrote: (01)
> On Thu, January 20, 2011 0:27, John F. Sowa said:
>> A question about types, sets, and classes arose on the AESIG
>> mailing list, which is related to the earlier discussion in
>> this thread. See, in particular, the slides by Peter Aczel.
>> -------- Original Message --------
>> Subject: Re: [architecture-strategy] Relationship between types, classes
>> and sets
>> Date: Wed, 19 Jan 2011 23:54:43 -0500
>> From: John F. Sowa <sowa@xxxxxxxxxxx>
>> To: architecture-strategy@xxxxxxxxxxxxxxx
>>> The terms "type", "class"ˇ and "set"ˇ are used within many
>>> languages, formal languages and natural language. A precise
>>> specification of languages involving these terms must have them
>>> precisely specified.
>>> Rick murphy referenced this paper:
>> These slides by Peter Aczel distinguish the terms 'type', 'class',
>> and 'set' as used by logicians who talk about higher orders of
>> infinity. Most of that discussion is irrelevant to AESIG.
>> 1. A set is extensional: it is uniquely determined by its elements.
>> 2. Type is an informal notion has been formalized in different ways.
>> But a very common and useful way is to choose some predicate
>> that specifies the type.
>> 3. A class is the extension of some predicate.
>> 4. Every set is a class, but some classes could be too big to
>> be sets -- but those are hyper-infinite monsters that are
>> irrelevant to computer systems.
>>> If an object could change types...
>> No object can "change" types without becoming a different object.
> I think that John refers to a mathematical object here.
>> [Example of type Integer vs. floating point number]
> However, for a temporal object, such as a person, the specific "type" of
> that object can change. (02)
Really, that depends on how you approach an ontology of times and properties.
There are several ways to do it, and what you say is correct in some but
strictly incorrect in others. IN a so-called "4-D" approach, one would speak of
temporal 'slices' of Barak Obama and predicate properties of them. They, of
course, do not change: the intuitive notion of 'change' is coded in such an
ontology as two different time-slices having different properties, rather than
one and the same thing having different properties at different times. Both
ways of ontologizing change have their merits and downsides, and both have
their passionate followers. But none of this has anything directly to do with
classes or sets. (03)
> For example, at one time Barak Obama was of type
> HumanChild, while at another he was of type HumanAdult. Of course, there
> is some more generic type of which the object is an instance throughout
> its existence. But sets, classes, and types can be generated or defined
> using narrower predicates.
> A standard distinction between a set and a class, is that membership in
> a class cannot change, while membership in a class can. (04)
That is not 'standard' in any formalism or school of thought that I am familiar
with. It is most certainly not correct if the class/set distinction is
understood in the sense used in NBG set theory, which has absolutely nothing to
do with time and change. It also is not correct if 'class' is understood in the
sense in which it is used in OWL and other, similar, Krep languages. (05)
> The set is either
> defined extensionally or generated as the extension of a predicate in a
> given context. Once the set is generated, the extension of the predicate
> in a different context (which might merely mean a different time) is no
> longer necessarily the same set. (06)
I see what you mean, but this notion is not 'standard' anywhere I know about.
To work it out in detail would require extending the semantics (and probably
the syntax) of extent formalisms in new ways, probably using ideas from modal
and hybrid logics. If you know of such work, I would be interested to hear of
Pat Hayes (08)
> If the extension of the predicate is context-independent, membership in
> the associated class is fixed. An instance of a context-free type (such
> as Integer) can not change whether or not it is an instance of that type.
> -- doug f
> doug foxvog doug@xxxxxxxxxx http://ProgressiveAustin.org
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