>> but I do not share the feeling that the first one (the negation) is really
>> compulsary.
>> [...]
>> For instance, I could defend a weakest form where 'some way' would mean:
>> plausibility(A given B) + plausibility(notA given B) ¾ 1
>
>The second-to-last character in the last line above, just before the final
>"1",
>comes across as "3/4" on my mail reader. What symbol was it supposed to be?
It says
plausibility(A given B) + plausibility(notA given B) <= 1
>Your belief in A increases, but you don't think your belief in (not A) should
>decrease?
In belief functions you have belief in A, belief in not-A, and belief in <A
or not-A>. This latter represents incompleteness in your knowledge. So you
might believe lung-cancer (.000001), not-lung-cancer (.95), don't-know
.049999.
There is also a plausibility measure that is equal to your belief in A plus
your uncommitted belief, so the plausibility is .05 for lung-cancer and
.999999 for not-lung-cancer.
Upon learning about the cough you might, for example, increase your belief
in lung cancer to .00001 and decrease your uncommmitted belief to .04999,
without changing your belief in not-A. This now makes lung cancer have
belief .00001; plausibility remains at .05. Belief in not-lung-cancer
remains at .95, while plausibility has decreased to .99999.
The point is that additivity does not necessarily hold because some of your
belief may remain "uncommitted."
> such a position seems
>inconsistent with deductive logic, and any system for plausible reasoning
>should
>contain deductive logic as a special case (where there is no uncertainty).
I'm not sure what you mean here. Both belief functions and probabily are
extensions to deductive logic that allow truth values other than zero, one,
or unknown.
In deductive logic all beliefs are zero (disproven), one (proven) or
uncommitted (neither proven nor disproven). Beliefs can change from
uncommitted to either proven or disproven, but can't go the other way. For
a proposition A that has been neither proven nor disproven, in what sense
would you say the belief in A and the belief in not-A must add to 1?
A belief function person would model an unproven proposition A as having
all its belief on the universal set <A or not-A>. If A is then proven, the
belief in A goes from zero to 1, the belief in not-A remains 0, and the
uncommitted belief goes from 1 to zero.
At this point, let me take time out to declare that I am NOT a proponent of
belief functions. I understand the discomfort that many people have with
the additivity axiom, but I myself prefer higher-order probability. A
model with higher-order probability can always be "collapsed" to a point
probability, and therefore the additivity axiom is satisfied, but this
point probability is highly uncertain, and can change dramatically as new
evidence is observed about the phenomenon.
The reason I am uncomfortable with belief functions is that I am not
convinced that the way beliefs are updated with new evidence (Dempster
conditioning, or its special case Dempster's Rule) is justifiable.
Attempts to develop interval probability calculi with different
conditioning rules tend to lead to intervals that are too wide and too
difficult to narrow with evidence. I remain open to a good argument for
Dempster conditioning or to some jusitifiable conditioning rule that gives
useful bounds. But I remain a skeptic.
>... To increase your belief in A without decreasing
>your belief in (not A) could only be justified if you were willing to
>admit the
>possibility that both A and (not A) could be simultaneously true,
Not true, as discussed above.
The belief function people remain firmly in the camp of those who accept
the Law of the Excluded Middle, at least on problems to which they apply
belief functions. It is the fuzzy set people who reject the exclusivity of
A and not-A. They say for "fuzzy" as opposed to "crisp" sets, the Law of
Excluded Middle may not hold. For example, a person cannot be both 2.32
meters and 2.33 meters tall. This is because height in meters represents a
crisp set. But a person who is 2.32 meters tall can be to some degree both
"tall" and "not tall," because tallness represents a set with fuzzy
boundaries.
There are people who are working on a decision theoretic semantics for
fuzzy sets.
Here too, I am in sympathy with those who are uncomfortable representing
all propositions as crisp sets, but I personally am not comfortable with
the approach the fuzzy set advocates take to modeling it.
Kathy Laskey