Re: looking for a predicate hierarchy

"Dmitry A. Kazakov" <mailbox@xxxxxxxxxxxxxxxxx> wrote in
news:9n4itzdwg52n.8juaqnsmr8vg.dlg@xxxxxxxxxx:

On Thu, 28 Dec 2006 18:50:21 +0100 (CET), V.J. Kumar wrote:


IVL also has a host of philosophical issues like how does
one substantiate *two* fuzzy membership functions ?

That's easy. You have a [fuzzy] subset A of some universe U and a set
of "focal" elements X={xi}, normally crisp, xi/\xj=Ø, \/xi=U. Each
focal element is a crisp subset of the universe. Let you have some
set-measure s. The upper set of A is a fuzzy subset of X such that
s(A/\xi). The lower set is s(~A/\xi). Now a hobby philosopher would
say, the universe can be sensed in terms of only X. Let us forget
about the nature of xi, which cannot be studied, and accept IFS as
"facts."

Well, it's too a strong "let us". The usual philosophical objection to
the standard FL membership function was its crispness. Say you have a
statement like Red(x) which is evaluated by an expert to e.g. 0.9. Then
it's natural to think that the statement itself is vague with a degree of
vagueness e.g. 0.7, then the statement about the statement about the
statement about the vagueness is vague itself, and so on. So you get an
infinite chain of vague statements and cannot in principle reason about
the fuzzy truth. This idea usually goes under the name of "higher-order
vagueness".

The FL folks answer to that is, ok, what we had until now was "type-1"
membership function, F1:X->[0,1] where X is some set, henceforth we'll
use "type-2" membership, F2:X->[0,1]^[0,1]. It turns out that using type-
2 FL is computationally infeasible, so the FL people use greatly
simplified interval-valued fuzzy logic, which is sometimes misnamed as
'intuitionistic' FL, instead of 'real' type-2 FL. So now we have two
simplifications, and it's unclear whether or not such simplification is
legitimate.


How did you arrive at the crisp value of 0.9 or 0.6 in your fuzzy
system ? ;)

A good question. The answer is that they are need not to be. Here I
mean intuitionistic fuzzy logic based on a four-valued one. A pure
fuzzy logic based on [0,1] has no satisfactory answer to your
question. But with four values as the bounds the answer is that 0.9
and 0.6 are estimations.

So how does one arrive at the estimations?

Through inference rules from "fuzzy facts."

Here's an inference for you from "fuzzy facts" courtesy of Edgington ;)

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.


Now, you have to
substantiate *two* fuzzy interval boundaries intead of one fuzzy
number. It's hardly better.

It is better because it handles uncertainty and contradiction. One
number, or anything else with an order cannot do that. The reason is
same as why we go four-valued.

No, a implies b equals 1 if a in {0, _|_} and b otherwise:

| b T 0 1 _|_
a |
---+-----------------
T | T 0 1 _|_
0 | 1 1 1 1
1 | T 0 1 _|_
_|_| 1 1 1 1

T implies T evaluates to T which prevents explosion with any
arbitrary formula.

I see.

[...]
So the idea with defining the implication is to prevent explosion
which is ensured by T a> T evaluating to T, and the rest of the
table is cooked so that MP would work.

But it does not!

(A /\ (A a> B)) a> B

evaluates T in A=T, B=T and in A=1, B=T.

But that's OK because T being a designated truth value means that the
formula holds ("has a model") !

T is "neither," it is "closer" to 0. It seems that it actually was:

a>| T 0 1 _|_
---+-----------------
T | 1 1 1 1
0 | 1 1 1 1
1 | T 0 1 _|_
_|_| T 0 1 _|_


No it was not, it was exacly as I specified.


Usually inference should be made only under certain truth. I.e. when x
a> y does not evaluate 1, then x|=y is wrong.

If you think so, then you've just destroyed the whole area of
paraconsistent logic, perhaps deservedly, but that's another question
;) To treat 'T' as a designated truth value is exactly what various
paraconsistent logics do to avoid explosion.

.

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