On Mon, May 7, 2012 21:47, John F Sowa wrote:
> Doug and Jack, (01)
> Before commenting on your comments, I'd like to repeat the
> point that these are two "short and general" definitions of
> the way the words 'class', 'type', and 'set' are used in most
> discussions the relate all three. (02)
I am used to discussions that distinguish 'class' from 'set' because
a class in general can have different elements at different times,
while a set can not. (03)
> But there are many different schools of thought that use and
> develop those notions much further. In most cases, their
> definitions are consistent extensions or specializations of
> the two short definitions below. (04)
>>> My recommendation is to use the following definitions:
>>> 1. A class is a set of all x of a given type.
>>> 2. Every type can be specified by a monadic predicate that is
>>> true of every instance of that type.
>>> These two definitions are sufficiently short and general that they can
>>> be specialized to all the major notations and schools of thought. (05)
>> This raises the question of what to do about Xes that are sometimes
>> of one type and sometimes of another. At any given time (and context)
>> there is a set of all X of a given type. However, at another time (or
>> context) there may be a different set of all Y of that same type. (06)
> As I said, those definitions are sufficiently general that they are
> consistent with any such extension. For example,... the class
> of real numbers and the class of rational numbers. (07)
These are classes of atemporal things and do not change membership.
So they can be mapped to intensional sets. (08)
> The question about times and sets raises other issues. Since the
> identity conditions for sets are determined by their instances,
> you can't change the instances of a set without causing it to become
> a different set. (09)
That's the problem. A class such as Adult, Ice, Butterfly, Tax, ...
has different members at different times, so can not be a set. Some
people get around this by making these attributes of an object, not
classes of object. This is a different definition of class from what I
take to be the standard definition. (010)
You define a class as "the set of all instances of a given type". In
order for a definition of a set to identify a timeless group of things,
the definition must yield the same group no matter when its extent
is calculated. This could happen if the definition were made timeless,
e.g., "the set of all instances of a given type at a given time" or "the set
of all instances of a given type at any time", or if the property of being
an instance of a given type, was a timeless property. (011)
You don't define "type" above, just specifying a property it holds,
so it is unclear whether being an instance of a given type is a
timeless property or not. You state that "[e]very type can be specified
by a monadic predicate that is true of every instance of that type",
but you don't state that "every monadic predicate specifies a type that
has as instances, exactly those entities for which the predicate is true".
Many monadic predicates have different extents at different times --
which would mean that if they were used for defining a type, the type
would have different instances at different times. (012)
A class such as Person, Planet, or Country arguably also has different
members at different times. However, some philosophers (especially
four-dimensionalists) would claim that the membership of such classes
does not change, merely the state of existence of its members change. (013)
> In programming languages, you can talk about the value of a variable
> at different times. The variable x can denote different sets at
> different times, but the sets themselves don't change. The only
> thing that changes is the denotation of the variable x. (014)
Are you suggesting that
"A class is a set of all x of a given type."
is to be interpreted as
"A class is what is referenced by a pointer to the set of all x of
a given type at the time for which the class is referenced"? (016)
> In any case, none of these issues require any change to the above
> definitions. (017)
I agree that the cases you mention do not. (018)
Do you consider that Adult(x) defines a class, Adult? If so, is
the class Adult a set? If so, what are its members? (019)
If you do not consider that Adult could be a class, do you
consider Person(x) to define a class, Person? If so, did
the class Person have any members 5,000,000 years ago?
-- are they the same members that it has now? (020)
>> To me this says that 'ontologists' do not discern content and
>> structure (endogenous attributes) from behavior (exogenous
>> attributes). Am I understanding your intent?
> Again, my only intent was to state definitions of the common ways
> that three words are used: 'class', 'type', and 'set'.
> But the distinction you're making about endogenous and exogenous
> attributes have been made by ontologists since Plato and Aristotle.
> In Aristotelian terms, the essence of something is determined by
> its *substantial form* which doesn't change as long as it exists.
> For a cat or horse, the substantial form corresponds to its DNA,
> which controls its growth from fertilized egg to death. You could
> call everything determined by the DNA its endogenous attributes,
> but that is not a term that Aristotle used. (Actually, he didn't
> use the term DNA, but if he had known about DNA, he would probably
> have said that the DNA is part of the essence of a living thing.)
> For what you're calling exogenous attributes, Aristotle used
> the term 'accident'. In his ontology, he assumed 9 kinds of
> accidents: Relation, Quantity, Quality, Active, Passive,
> Condition, Position, Time, and Place.
> A cat or horse can't change its DNA, but any or all of the
> accidents can change at any time.
> In any case, all of these issues are compatible with the
> definitions I suggested for the terms 'class' and 'type'.
> And these are not definitions I invented. They or something
> equivalent are commonly used. (021)
Even if Class meant Aristotelian essence, that essence stops existing
when the holder of that essence does. In order for your definitions
to be timeless, it seems that you must restrict "type" to the timeless
attribute of something having an essential property at the time it
existed/exists/will exist. Is this your meaning of class? (022)
-- doug (023)
> John (024)
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