[om-list] Sense & Evaluation

[Mark Butler]

In regard to common data models, I have a few comments.   In my current 
system, I do not store attribute values in the attribute objects, but 
rather in evaluation objects.  Any person can have their opinion about 
any concept, attribute, or relation expressed in an evaluation object.

My evaluation object consists of the following:

subject          person / entity who made this evaluation
object           The concept, attribute, relation being evaluated

sense            subject's sense from -infinity (denial) to
                 +infinity (absolute certainty), zero neutral
minimum value    subject believes value greater than this
                 (e.g. event date after 1789)  [sense positive]
expected value   subject expects value to be this (e.g. 1791) [sense 
positive]
maximum value    subject beleives value  less than  this  (e.g. 1803) 
[sense positive]


This is very simple, but probably adequate to represent someone's 
opinion about a historical scalar attribute or something else that 
cannot change.  The sense modifier is my favorite - by setting it 
negative instead of positive, the sense of the minimum, expected, and 
maximum values is reversed, just like throwing a "not" in front of a 
sentence.  In this example it would be like changing "event date was 
between 1789 and 1803" to "event date was not between 1789 and 1803".  

Sense could also be used to express denial of the existence of an 
object, by attaching the evaluation record to a regular concept instead 
of an attribute.  For example, given the concept "Unicorn",  sense 
positive would indicate belief in unicorns, zero would be non-committal, 
and sense negative would indicate disbelief.  I am thoroughly convinced 
using bipolar senses this way is far more user friendly than using [0,1] 
bounded unipolar fuzzy logic.

Standard probability theory only works well with boolean logic - as soon 
as you mix probability and fuzzy logic bad things start to happen unless 
you are extremely careful.   Mathematicians mostly solve the problem by 
avoiding fuzzy logic completely, but in natural language we do not have 
that option. 

[0,1] bounded fuzzy logic is based on the concept of fractional set 
membership.  Unfortunately, a fractional set membership value only 
converts well to a truth value for simple objects in the presence of 
perfect information.  Most real world objects are complex.  If you have 
a complex object that can overlaps large regions of a set,  you have to 
integrate the multiplication of the object function and the set function 
and normalize to get some sort of degree of containment.  Normalized, 
zero indicates zero overlap, and one indicates perfect containment of 
the object by the set.

If you have perfect information, you can do that integral nicely, of 
course.  But what if you know nothing about the object?  Should you just 
guess set membership 1/2 in desparation? But after the fact how do you 
tell the difference between actual set membership 1/2 (partial overlap) 
and estimated set membership 1/2 (unknown overlap)?  Saying 1/2 makes it 
appear like you know something when you don't.

One very painful way is to keep track of a membership probability 
distribution function.  That is the mathematician's answer.  Natural 
language doesn't work that way - instead we keep track of sense or 
agreement relative to propositions.  The nice part about sense is that 
it jointly represents ignorance, degree of belief, and fractional set 
membership very nicely.  If someone says "the object is a member of the 
set" - I can assent, dissent, or say nothing, which are positive, 
negative, and zero senses, respectively.

If I know nothing, my sense is neutral (zero).  If I have perfect 
knowledge that the set completely contains the object, my sense is 
certain (large and positive).  If I have perfect knowledge that object 
lies completely without the set my sense is denial (large and 
negative).  If I have perfect knowledge, I can get an actual degree of 
overlap, likewise if I have partial knowledge, I can get an estimated 
degree of overlap.

This doesn't solve the ambiguity problem, but it is a major improvement, 
one that meshes well with the way we talk about natural language and one 
that leads to simpler mathematical formulas. I suspect that [0,1] fuzzy 
logic can be shown to be equivalent to bipolar [+/-] fuzzy logic, but 
for me the latter is far more natural, particularly because it doesn't 
have this funny 1/2 floating around everywhere.

In bipolar logic you have two basic choices [-1,+1] linear logic and 
[-infinity,+infinity] logarithmic logic, where the left endpoint 
represents absolute denial, and the right endpoint represents absolute 
certainty. I prefer the latter because you can add instead of 
multiplying when doing most evidence calculations.  Logarithmic measures 
are widely used in science and engineering for the similar reasons.

If you have a proposition in log10 bipolar fuzzy logic, you get the 
following (roughly):

sense      zero-based certainty
---------  --------------------
-infinity  0        0
-6         10^-6   (0.000001)
-5         10^-5   (0.00001)
-4         10^-4   (0.0001)
-3         10^-3   (0.001)
-2         10^-2   (0.01)
-1         10^-1   (0.1)
 0         1/2     (0.5)
 1         1-10^1  (.9)
 2         1-10^-2 (.99)
 3         1-10^-3 (.999)
 4         1-10^-4 (.9999)
 5         1-10^-5 (.99999)
 6         1-10^-6 (.999999)
+infinity  1        1


This works very nicely for analysis of inductive propositions - as 
evidence is gathered for a scientific theory, for example, the certainty 
approaches one, but never reaches it because of the limitations of 
induction. However, the logarithmic sense measure keeps increasing 
linearly with the amount of evidence. 

Arguably, people do this conversion informally all the time - add up the 
amounts of confirming and dis-confirming evidence, and then convert to a 
linear bounded degree of belief between assent and denial.  A tiny 
amount of confirming evidence is no big deal, but a tiny amount of 
dis-confirming evidence is rather unsettling, especially since most 
people tend to think in boolean terms.

Have I made my case?

 - Mark










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