Re: looking for a predicate hierarchy
- From: "Dmitry A. Kazakov" <mailbox@xxxxxxxxxxxxxxxxx>
- Date: Mon, 1 Jan 2007 14:08:05 +0100
On Sat, 30 Dec 2006 20:09:45 +0100 (CET), V.J. Kumar wrote:
"Dmitry A. Kazakov" <mailbox@xxxxxxxxxxxxxxxxx> wrote in
news:1iijo50am364r$.bfl8qj4qnh2o.dlg@xxxxxxxxxx:
On Fri, 29 Dec 2006 20:40:20 +0100 (CET), V.J. Kumar wrote:
Anyway both solve the issue of "crispness." You have an interval (or
other compact set) of possible values and whether the bounds were
crisp plays no role.
How come ? If you use the interval simplification for higher order
vagueness which questionable in itself, then such simplification is
mathematically equivalent to Atanassov's idea of using two type-1
membership functions with crisp truth grades. So what's the gain ? Instead
of one dubious MF now you need to substantiate two !
These are only bounds, they need not to be exact. You can always say that
if x is not contradictory then it is in [0,1]. Do you need to substantiate
0 and 1?
Let x, y, z be three balls that an 'expert' determined to be red to
some degree and small to some other degree:
v(Red(x)) = 1 v(Small(x)) = 0.5
v(Red(y)) = 0.5 v(Small(y)) = 0.5
v(Red(z)) = 0.5 v(Small(z)) = 0
Now using Zadeh blessed definition for 'and' as min(x, y), we'll get
the conclusion that all the balls are equally red and small ! 'Red
and Small' equals 0.5 in all the cases which clearly contradicts the
intuition that x being red and small has to have a higher degree of
truth than y, and z has to have the lowest.
1. For z it is 0.
I was in too much of a hurry and mixed up OR and AND examples. The above
should have been:
1. 'X is red AND small' should intuitively have higher degree of truth than
'y is red AND small' but both evaluate to 0.5
2. Given Zadeh's definition of OR as max, 'y is red OR small' should
intuitively have a higher degree of truth than 'z red OR small' yet both
evaluate to 0.5
2. Why x should be more of Red/\Small than y?
I think it's obvious that while being of the 'same' size x is redder than
y. Any other interpretation of AND would be rather meaningless, or useless
or both. The same goes for the second example.
This is all about the set measure, which determines how the
conditional (Red/\Small | x) were related to (Red | x) and (Small |
x).
Now, that is a strange argument. The crucial difference between logic and
the probability theory is that the latter is not truth-functional as you've
mentioned earlier yourself. So it is not "all about the set measure" but
rather about whether or not FL should be non-truth-functional as much as
the PT is. Is it what you are saying ?
When you say that x is [intuitively] more red and small than y. That
assumes some sort of additive measure of Red/\Small.
Not every measure is additive. Also, if you think that additive measure is
universally intuitive, then consider obvious:
p(x is Red) /\ p(x is Red) > p(x is Red)
Is it? Does repeating a wrong statement make it more valid? That would be
rather a propaganda, than logic. (:-))
Note that I don't argue either against or for an additive measure. Further
an inability to build a logic in terms of /\ and \/ based on an additive
measure were not a valid argument against it. My position is that seemingly
logic with all its features plays a subordinate role to the set theory and
a measure there. If the measure is possibility (max), then the result is
intuitionistic fuzzy logic, when it is probability (sum), the result is
different and weaker. Both are *legal*.
If so than FL is not a logic
because truth functionality is one of cornerstones of anything that claims
to be a logic. Besides, I am not aware of any FL theorist who would claim
that. Edginton by the way using her examples does claim that the vagueness
theory should be non-truth-functional but she is not a FL theorist. So if
you have similar views you should abandon FL and use something else in its
stead ;)
Parenthetically, the probability measure by itself is not sufficient to
calculate conditional probability either. If it were, probabilities
composition would be truth-functional as well.
Zadeh system is obtained when the membership function of a set A
in the element x, A(x) were defined as pos(A|{x}).
I am sorry I do not understand that.
Let A:X->[0,1], B:X->[0,1]
pos(A|B) =def= Sup min{A(x), B(x)}
nec(A|B) =def= 1-pos(~A|B) = Inf max{A(x),1-B(x)}
From that nec(A|{x}) = pos(A|{x} = A(x), which gives the "meaning" of themembership function value of A in x. A(x) is the possibility and necessity
of A under the condition {x}.
Hmm, without going into philosophical issues about merits of
contradictory inference (not to be mixed with inference from
contradiction), but purely technically, less inference paths you take,
smaller is the set of consequences. So inference under certainty
cannot explode more than one under certainty + contradiction.
Could you explain ? I am not sure what you mean by that. An example would
be nice.
Consider the implication you referred.
a> | T 0 1 _|_
---+------------------
T | T 0 1 _|_
0 | 1 1 1 1
1 | T 0 1 _|_
_|_| 1 1 1 1
When reasoning under certain truth, the only valid paths were 1=(x a> y):
a> | T 0 1 _|_
---+------------------
T | - - 1 -
0 | 1 1 1 1
1 | - - 1 -
_|_| 1 1 1 1
For reasoning under contradiction or else certain truth [ 1=(x a> y) or
T=(x a> y): ] it would be:
a> | T 0 1 _|_
---+-------------
T | T - 1 -
0 | 1 1 1 1
1 | T - 1 -
_|_| 1 1 1 1
Here we can deduce more: {T, T} |= (T=(T a> T)) and {1, T} |= (T=(1 a> T)).
But this cannot influence explosiveness of inference under 1=(x a> y), in
the sense, that if 1=(x a> y) were explosive then 1=(x a> y) or T=(x a> y)
would be as well.
--
Regards,
Dmitry A. Kazakov
http://www.dmitry-kazakov.de
.
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