Re: looking for a predicate hierarchy

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

"Dmitry A. Kazakov" <mailbox@xxxxxxxxxxxxxxxxx> wrote in
news:o4bdwl2r39pm.33vcggeeyv1a$.dlg@xxxxxxxxxx:

On Wed, 27 Dec 2006 20:10:48 +0100 (CET), V.J. Kumar wrote:

"Dmitry A. Kazakov" <mailbox@xxxxxxxxxxxxxxxxx> wrote in
news:av8llm95a8p5$.jjkecdqw8xub$.dlg@xxxxxxxxxx:

On Tue, 26 Dec 2006 19:38:29 +0100 (CET), V.J. Kumar wrote:

Right, but we have to loose that anyway. To me four-valued logic is
just a necessary step to a fuzzy one. There you will never have a
chance to save it.

Well, not quite. It depends what you mean by fuzzy logic. There
are quite a few of those. The most known varieties like
Lukasiewicz's, Godel's, product logic, all have the fuzzy
implication connective, deduction system which is purely
syntactical, are sound and complete (see Petr Hajek's online
articles for an introduction, e.g. Basic Fuzzy Logic and
BL-algebra", or his book "Metamathematics of Fuzzy Logic").

These are systems which don't stand the question you pose later. The
only consistent fuzzy logic can be based on the possibility as the set
measure. That inevitably leads to intuitionistic fuzzy logic with
Belnap's four values as the bounds.

It's unclear what you mean by "intuitionistic fuzzy logic". Is it
Takeuti's logic or Atanassov's ? If it's the latter,

Yes

"intuitionistic"
is a misnomer because it has been shown that, although based on
different insights, IFL is mathematically equivalent to interval-valued
fuzzy logic.

Surely they are. However, IFL and IVL were introduced as continuations of
three state logic based on 0, 1, _|_. When I consider IFL, I would also
allow contradiction, i.e. intervals with the lower bound greater than the
upper one. That would be a continuation of a four-valued logic.
Contradiction is needed for many reasons.

IVL laso 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."

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."

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 _|_

Or did I misunderstand you ?

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

Arguably one could consider the case when it evaluates _|_ (both true and
false) as both x|=y and not(x|=y), and then continue inference along both
paths. However, because not(x|=y) ["x does not imply y"] would be a quite
weak statement in any four-valued logics, it might deliver nothing useful.
In any case we cannot just consider 1 and _|_ same in inference, if we
haven't proved that all possible consequences were indeed same. With a>
they don't look same! The two lower rows of the truth table aren't
identical. Then I have a vague suspicious that this would trivialize logic.

Having said that, I actually very like the idea to use all possible [four
and more in IFL] outcomes of evaluation of x=>y. That means that "imply"
would have no any special meaning, just some relation between x and y. I
think it would be a natural consequence of four-valued approach.

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