Re: looking for a predicate hierarchy

On Tue, 2 Jan 2007 15:49:46 +0100 (CET), V.J. Kumar wrote:

"Dmitry A. Kazakov" <mailbox@xxxxxxxxxxxxxxxxx> wrote in
news:1d4hajhvtd9rg$.1r3hyvmtauslj$.dlg@xxxxxxxxxx:

On Mon, 1 Jan 2007 18:35:22 +0100 (CET), V.J. Kumar wrote:

"Dmitry A. Kazakov" <mailbox@xxxxxxxxxxxxxxxxx> wrote in
news:85gat69gqvtq$.11ps8kt0xar5e$.dlg@xxxxxxxxxx:

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?

You seem to have misunderstood. What I am saying is that the
interval is defined by two crisp real numbers that correspond to two
membership functions F- for the lower boundary and F+ for the upper
boundary according to Atanassov. I do not see why using two crisp
numbers like the judgement 'ball_is_red = [0.7..0.75]' is any better
than using one 'ball_is_red=0.7'.

No. It is

ball is red = 0.7 vs.
ball is red *in* [0.7..0.75]

That's what I said, the boundaries are still crisp, and that's the
problem no different from having a single number. How does one
substantiate the crisp boundaries ? Using an interval instead of a
single number is just an illusion of vagueness. As was said before,
choosing two crisp boundaries for the interval is equivalent to choosing
two crisp type-1 numbers.

It actually could be [0.73, 0.74] or [0.7, 0.71]. They don't even
intersect. Is that not vague enough?

The point is that equality = is replaced by inclusion, which moves
uncertainty from the value to the relation between "ball is red" and a
value of.

Instead of using one crisp number we use two so this way of doing
thing looks even more suspect than before.

It is only syntactically two numbers. Semantically it is an
uncountable set of numbers.

Right, but you've defined the set with two crisp numbers thus losing the
very vagueness you were trying to model.

Not at all, because the identity relation (=) was replaced with inclusion
one. That is a sufficiently different structure. In particular from x in
[0.7, 0.75] and y in [0.7, 0.75] does not follow x=y. Compare it with:
{A=_|_, B=_|_} does not imply that both A and B were either true or false.
I.e. A/=B.

First of all, arguments to "real world" are automatically disqualified
as philosophical. (:-))

I disagree unless you consider a fuzzy logic as a meningless but
entertaining symbolic game which is a perfectly legitimate point of view.

Huh, but you were arguing to intuition. Is intuition any real? What can be
said about the correspondence between "real intuition" and "real world"?

(Isn't it intuitively clear that the Earth is flat and Visual Basic is the
best programming language? (:-))

I just tried to model your notion of "better" choice. Additivity was a
formalization of having s(x in Red/\Small) > s(y in Red/\Small).

No, measure additivity is the property that for mutually disjoint subsets
S1 and S2, m(S1 U S2) = m(S1) + m(S2). It's not the 'and' connective,
it is 'or'. I am not sure why you are using the conjunction in your
example.

Because m(S1/\S2)=m(S1)*m(S2), when S1 and S2 are independent. So if
redness and smallness were mutually independent and m additive, then you'd
get the "intuitive" result m(Red/\Small|x) >= m(Red/\Small|y).

Additivity depends on the elementary events/outcomes which compose x,
y, Red, Small. Only when these elements are mutually independent we
might get something like an additive measure. The opposite situation
is when they are nested. In that case the measure would be max/min. In
"real world" you might have nether.

Not every measure is additive. Also, if you think that additive
measure is universally intuitive, then consider obvious:

In fact, with the above example, I used a non-additive measure.
According to Zadeh, the membership function *is* a possibility
distribution which is a pretty reasonable point of view as membership
degrees are clearly non-additive. Just to make sure we are on the
same page, in simple terms measure additivity is just the property
that m(A OR B) = m(A)+m(B) where A and B are some sets (see
Kolmogoroff axioms).

Yes, that is what I meant.

Clearly, the property has not been (and could not be) used in the
red balls example.

We need a clear model behind the example. Further, possibility can
always be used because pos(x in Red/\Small)<=min{pos(x in Red), pos(x
in Small)} is universally correct.

Why is it "universally correct"?

Because

Sup min{x(w), min(Red(w), Small(w)) <=
min {Sup min{x(w), Red(w)}, Sup min{x(w), Small(w)}}

For any x, Red, Small.

Zadeh sez that possibility
distribution is the same as the membership function, further he states
that v(X and Y) = min(v(X), v(Y)), where v is a membership degree or
*possibility*. It is not '<=', it is '='.

Yes, inequality appears when the conditions become non-singleton sets. For
elementary events:

pos(A/\B|{w}) = // (A/\B)(w)
= min{pos(A|{w}), pos(B|{w})} // min{A(w),B(w)}

In case of a general set C it is only:

pos(A/\B|C) <= min{pos(A|C), pos(B|C)}

Therefore when reasoning about A/\B not knowing the condition C, the best
we could say is that pos(A/\B)<=min{pos(A), pos(B)}

Granted, other folks have
different ideas on whether the possibility distribution and the
membership function are the same thing or not. Besides, I am not sure
why we are even talking about possibility theory which is a related to
but may be a different beast from the fuzzy logic as formulated by Zadeh.

Because you asked sometime before about the meaning of membership function
in a Zadeh-like system. The meaning is possibility. Further my point was
that when looking for a meaning, we should consider/invent the underlying
set measure. The measure is that meaning.

From that you can get that both

pos(x in Red/\Small) <= 0.5
pos(y in Red/\Small) <= 0.5

So? You are still free to hold x in Red/\Small for more possible, if
you could add your intuition as an evidence. That formally means, that
you possessed some additional knowledge about relations between x, y,
Red and Small. This might improve the estimations above.

Further, any upper bounds don't tell anything about the actual
relation between x and y being in Red/\Small, only about the
possibility of. You need lower bounds, the necessities, to make any
conclusions. But even then, you would have only two estimations of x >
y | Red/\Small, i.e. the upper and lower bounds for the possibility
and necessity that the measure of x is greater than the measure of y
on the set Red/\Small.

[This is exactly the same situation as with probabilities. Once you
get in, you can't obtain nothing but measures.]

I think we've already been through this. It cannot be exactly the same
just because any logic, including the original formultaion of the fuzzy
logic by Zadeh and others, has always been understood as being truth-
functional. What you wrote above rejects the truth-functionality and
disqualifies the system from being called 'a logic'. One has to appeal
to other factors, like the model structure, necessities, possibilities,
what not, in order to derive a conclusion. There is nothing wrong with
such appeal, but it simply means that the fuzzy logic by itself cannot
do what it was supposed to do, namely derive meaningful conclusions
based solely on the truth values of a compound statement constituent
terms.

Yes, I agree with your analysis. Unfortunately, but a logic with the
properties you desire just cannot be constructed. I think it should be
obvious that contradiction and uncertainty (be it randomness or fuzziness)
cannot be handled in a purely symbolical manner. The way uncertainty
propagates is determined by the structure of. You cannot ignore it without
some loss in the inference system.

Having said that, I also doubt that the classic logic in fact had all these
properties. The problem is that the system of equations

{A=a, B=x, (A=>B)=b}

cannot be solved analytically as:

x=f(a,b)

in that logic as well.

So implicitely, by appealing to a possibility theory, you admit that a
fuzzy logic is not an adequte tool even for a problem as simple as the
ball one ?

But this is not a simple problem. You want to draw some conclusion about
balls knowing nothing about balls their colors and sizes. Why should that
be doable? IFL just gives you estimations considering all possible
relations between balls, colors, sizes once in an optimistic and once in a
pessimistic way. You cannot ask for more, without providing a more detailed
description of these things.

Take another example, you neeed to hire someone as a lead Java
developer who knows both Java, naturally, and SQL. You have two
persons with (1, 0.5) and (0.5, 0.5) qualifications. Who are you
going to hire ? The fuzzy logic says it does not matter (see the
ball example). What measure/possibility distribution/membership
function would you come up with to solve the problem correctly?

OK, Let you were interested in a person who knows both Java and SQL.
That formally means that you want to know the measure of Java/\SQL
provided given person x, i.e.

pos (Java/\SQL|x)
nec (Java/\SQL|x)

This 'SQL|x' thingy makes your reasoning non-truth-functional exactly in
the same way as using the conditional probability notion. See also below.

Right, because as I said in previous posts, I believe that the approach is
basically same, with the only difference in the measure being assumed.

The latter tells us how it is possible that the person x would not
know Java and SQL:

nec (Java/\SQL|x) = 1 - pos (~(Java/\SQL)|x)

This model considers anything outside Java/\SQL as non-asset. Is that
really true? If yes, then the answer was correct. Because no matter
how much were Java/\~SQL|x, that would not be an asset.

But consider this:

pos (Can_learn_SQL_in_a_week /\ Java /\ ~SQL | x)

Does this matter to the choice? If yes, then Java/\SQL was a wrong
objective. Inadequate models provide inadequate answers.

This is all very nice, but you are trying to solve the problem with
tools *other* than a fuzzy logic where all you have is membership degrees
and rules to compose them.

If you mean classical FL with logical values from [0,1], then yes, I am.

Does it mean that a fuzzy logic is useless
and one has to move to more realistic theories like a possibility theory
or may be just use the old probability theory ?

I think that realistic were a mixture of. Probability theory fundamentally
cannot describe contradiction. Possibility is unable to adequately describe
randomness. In the "real world" we have both intermixed.

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.

No, exposiveness is defined as *all* the formulas hold as a
consequence of a certain formula/set of formulas. You've just
demontrated a valid derivation, not explosivity.

The point is why *more* valid derivations should make anything else
non-explosive?

A derivation makes a logic trivial/explosive only when you can derive
*all* the possible formulas from a certain formula, like from (x and ^x)
in the classical logic. If you can derive more (when you redefine the
implication), but not *all*, then the logic is not explosive.

This is OK, but it would not save an *already* exploding logic. Thus if a
logic is explosive under a certain implication it also will be under a less
certain one.

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

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