On 13 Sep., 19:44, Brent Meeker <meeke....domain.name.hidden> wrote:
> Youness Ayaita wrote:
> > ...
> > I see two perfectly equivalent ways to define a property. This is
> > somehow analogous to the mathematical definition of a function f: Of
> > course, in order to practically decide which image f(x) is assigned to
> > a preimage x, we usually must know a formula first. But the function f
> > is not changed if I do not consider the formula, but the whole set
> > {(x,f(x))} instead, where x runs over all preimages.
>
> > Concerning properties, we normally have some procedure to define which
> > imaginable thing has that property. But I can change my perspective
> > and think of the property as being the set of imaginable things having
> > the property. This is how David Lewis defines properties (e.g. in his
> > book "On the Plurality of Worlds").
>
> But I don't think you can define a property this way. For example,
> suppose you want to define "red". Conceptually it is the common
> property of all things that are red. But this set isn't given, and it
> can only be constructed (even in imagination) if you already know what
> "red" is. For a strictly finite set you could use ostensive definition
> to get the set, but I suspect you don't want to limit your set size.
>
> In any case I don't think "imaginable" and "describable in some
> alphabet" are equivalent. People construct perfectly grammatical noun
> clauses that don't correspond to anything imaginable, e.g. "quadratic
> chairs".
>
> Brent Meeker
I've already explained how my (or Lewis's) definition of a property is
to be understood correctly. Of course, practically I can only try to
construct the set of imaginable things that are red if I know a
procedure how to decide if something is red in every particular case.
But this is only related to the practical applicability of the
concept. We agree that the property "red" is completely defined by the
set of imaginable things being red. So, whenever it's useful, I may
work with this set instead of our common conception of "red" (I will
never have the concrete and full set at my disposition but that won't
be necessary). And you will se below why it is useful to do so.
Your second remark is very interesting. You're right that the English
language can construct difficult situations when it comes to
descriptions of possibly imaginable things. This is why I avoid the
English language in this context (even the French language, which is
said to be very exact, is not an option). Two ideas how to get the
Schmidhuber ensemble of descriptions out of the "set" of all
imaginable things:
1st idea:
Let T be the set of all imaginable things. Then, corresponding to my
definition of a property being a subset of the T, the power set P(T)
is the set of all properties. To describe an imaginable thing t, we
might proceed as follows:
For every property p in P(T), we say wheter t has the property (then
we assign a 1) or not (we assign a 0). The set of all descriptions
then is {0,1}^P(T) similar to the Schmidhuber ensemble. The only
problem with this is the cardinality of the ensemble. The construction
{0,1}^P(T) is equivalent to the power set P(P(T)). This means, if T
has the cardinality of the natural numbers, then P(T) has the
cardinality of the real numbers and P(P(T)) has an even higher
cardinality! Since the Schmidhuber ensemble only has the cardinality
of the real numbers, we're facing a problem at this point.
This leads to the
2nd idea:
We don't say that imaginable things are fundamental, but that the
properties themselves are. This idea was also expressed by 1Z in his
last reply ("We define imaginable things through hypothetical
combinations of properties", Z1) and I think it's a very good
candidate for a solution. Then, we start from S being the set of all
properties (perhaps with the cardinality of the natural numbers). As
above, we define {0,1}^S as the ensemble of descriptions. This would
have the cardinality of the real numbers and could mathematically be
captured by the infinite strings {0,1}^IN (the formal definition of
the Schmidhuber ensemble to give an answer for Bruno).
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Received on Thu Sep 13 2007 - 18:36:49 PDT