Re: looking for a predicate hierarchy

On Fri, 29 Dec 2006 20:40:20 +0100 (CET), V.J. Kumar wrote:

"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.

Well, integers are "simplified" reals. That alone does not devaluate
integers. IFS can be extended to type-2, but I don't think it would be
worth of efforts. Especially when it gets to specifying a distribution
along the Y-axis. That quickly becomes as empiric as statistic
distributions. In fact it would be interesting to mix IFS with probability
distributions to get random fuzzy sets.

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.

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.

1. For z it is 0.

2. Why x should be more of Red/\Small than y?

This is all about the set measure, which determines how the conditional
(Red/\Small | x) were related to (Red | x) and (Small | x). Zadeh system is
obtained when the membership function of a set A in the element x, A(x)
were defined as pos(A|{x}).

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.

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.

--
Regards,
Dmitry A. Kazakov
http://www.dmitry-kazakov.de
.

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