>Pat,
>
>> >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.
>
>That's correct.
>
>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.
>
>This raises some tricky mathematical issues. (01)
The math is straightforward. The trickiness lies in interpreting it
properly. One gets similar issues all over the place, eg moving
things are at a single spatial point at each point in time but are
also moving. This hasn't been mathematically a problem for several
centuries but it still regularly gets ontologyaxiomwriters into a
tangle. (02)
>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.
>
>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. (03)
Has anyone thought of what a statistical theory would look like if it
were built on a constructivist mathematical foundation? (I'm sure the
answer is Yes, but I have no idea where to look.) (04)
Pat (05)


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