Pat, (01)
> >What I was asking is how a language such as IKL, which is a
>>superset of FOL that also supports metalevel statements, could
>>be used to represent the kinds of operations required for
>>probability models.
>>
>>> What you describe is far too simplistic. It's nearly impossible
>>> to create a probability model that way that's not either utterly
>>> simplistic or inconsistent.
>>
>>I used a very simple example, but the IKL mechanisms can be used
>>to support metalevel statements about propositions, the structural
>>components of propositions, their relationships to numerical
>>values, and the operations on those values.
>
>IKL supplies natural numbers but it doesnt have builtin reals. So
>you will need some extra machinery of some kind to be able to do
>continuous math, for example for talking about distribution curves or
>integrals. (02)
That's correct. (03)
To do probability right, you need real numbers, and you need to be
able to do continuous distributions. For example, consider a number
picked at random between zero and one. (Imagine a spinner that can
land anywhere on the disk with equal likelihood; measure the distance
around the edge to where it landed and divide by the circumference 
the number you get is uniformly distributed between zero and one.)
The probability of getting a number in any subinterval of the unit
interval is equal to the length of the subinterval. But the
probability of getting any specific number is zero. (04)
This raises some tricky mathematical issues. In finite domains, we
can equate probability zero with unsatisfiability and probability 1
with validity. But in the example I gave above, UniformRand = X is
satisfiable for any X between zero and one, yet (Prob (UniformRand =
X)) is zero for all X, and moreover, Prob(Exists X UniformRand=X))
is equal to 1. (05)
This problem comes up in countably infinite domains, too. For
example, consider infinitely many tosses of a fair coin. Any
sequence of heads and tails is possible, but each sequence has
probability zero. To be consistent with the statistical theory, the
sequence has to have 50% heads and 50% tails (to be precise, the
limit as n approaches infinity of the number of heads in the first n
tosses divided by n has to be 0.5). Sequences with a different
limiting proportion, or with no limiting proportion, are inconsistent
with the statistical theory. (06)
> >> Over the past several decades, statisticians and computer
>>> scientists have learned a great deal about how to represent
>>> probabilistic knowledge.
>>
> >I'm sure they have, but the IKL mechanisms can support those
>>representations. Anything that can be defined in PROWL or
> >BayesOWL can be defined in IKL plus much, much more. Numerical
>>functions of any kind
>
>Any kind?? (07)
If we restrict ourselves to IKL functions that can be represented
using a finite number of bits, there are only countably many of them.
There are uncountably many probability distributions on the integers. (08)
Kathy (09)
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