Re: math development curiosity question

In article <1144960084.171394.17920@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
david petry <david_lawrence_petry@xxxxxxxxx> wrote:

Herman Rubin wrote:

david petry wrote:

When we compute, we use numbers with a finite precision.

I know of NO theory of finite precision reals. Any
approach which does not have (x+y)-y = x EXACTLY will
not lead to a theory.

What I have been arguing is that we use probabilistic logic for
reasoning about finite precision reals, and then to recover classical
logic and classical reals, we take the limits as the uncertainty goes
to zero. We could call that a "theory", and it doesn't require
anything "EXACTLY".

The justification for using probabilistic logic is this: uncertainty is
something which simply must be dealt with when reasoning about the real
world, so if logic is to provide us with the tools to reason about the
real world, it must deal with uncertainty. Formal, exact logic is an
abstraction derived from the logic that deals with uncertainty.

The discussion of probability uses Boolean logic. There
are other Boolean algebras than the 2-element one, and one
of these consists of the measurable sets of a probability
space, or those modulo set of measure 0. However,
probability is NOT a truth value, as a truth value system
has the truth value of the result of a logical operation a
function of the truth values of the arguments.

Now suppose, P(A) = .5; then P(~A) = .5. But P(A&A) = .5,
while P(A&~A) = 0. This difficulty arises with all attempts
to have a linear truth-value system with more than two values.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@xxxxxxxxxxxxxxx Phone: (765)494-6054 FAX: (765)494-0558
.

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