Re: looking for a predicate hierarchy

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

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?

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


When you say that x is [intuitively] more red and small than y. That
assumes some sort of additive measure of Red/\Small.

It does not assume anything of the kind. I think you've got it backwards.
We want to model some fragment of the 'real world' so we look for some
tools, not the other way around, we take a tool and try to make reality
fit the tool, be it fuzzy logic, possibility theory, or whatever. With the
tools at our disposal, one would expect that x being red and small is a
better choice than y, if the tools say 'no', than they are not good
tools.



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


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. (:-))

I do not see the point. According to the red balls example, p(x is Red) /
\ p(x is Red) = p(x is Red) = 0.5


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.

See above.

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
the membership function value of A in x. A(x) is the possibility and
necessity of A under the condition {x}.

Well, no, Zadeh just equated the membership function and the possibility
distribution function. Maybe he modified his position later, I do not
know. Dubois, the one who established IVL and Atanassov's IFL
equivalence, somewhat differed wrt to possibility distribution. In any
case, its unclear how it helps to repair the problem with the red balls
example.


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?


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.


.