Re: looking for a predicate hierarchy

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. That is my view on the things. There could be
alternatives based on other measures, like probability (Pr), for instance,
but they usually cannot produce a logic in the sense that V and /\ cannot
be defined as functions of the arguments. Pr (A U B) /= Pr (A) + Pr (B),
only when A and B don't intersect.

It's a controversial issue but I find FL account of vagueness simplistic
and not convincing, for example how does one determine the value of
membership function for borderline cases; combining fuzzy truth values
seems naive and so does FL's truth functionality as whole; at some point
one has to make a decision whether or not to do something in which case
logic collapses to binary, et).

Defuzification is outside FL. That's no different from "de-randomization"
in probability theory. (You have x probability of shock reading
comp.object. Would you read it here and now?) Fundamentally, if we could
deduce from fuzzy anything but fuzzy, then that would
kill/explain/trivialize fuzzy. Applicability of fuzzy values (as well as
their sources) is not an issue and the very question is wrong (outside FL).

You may want to read Parikh's "Test for
Fuzzy Logic" and Hajek's opposing view in "Ten Questions on Fuzzy Logic"
where Hajek himself states that FL gives relative, not asolute conclusions
(comparative truth).

When it rains to 0.9 degree and does not to 0.6,
then what?

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. The
inference does not produce crisp values. It produces crisp estimations of.
I.e. pos(A)>= 0.9, nec(A)<=0.6 (equivalently pos(~A)>=0.4). This eliminates
the apparent contradiction.

The disjunctive syllogism will hold for the formula (A=>A). Just
substitute (A->A) for the letter and follow the reasoning.

But

((A or B) and not A) => B

is not universally true (for example in A=T, B=0). And we cannot use
-> and => interchangeable. One should stick to one of them.

It's a typo: I should have written just substitute (A=>A) for ...

OK, for =>, when A=T and B=0 the above becomes:

((T V 0) /\ not T) => 0 =
(T /\ T) => 0 =
T => 0 =
_|_

i.e. it does not hold.

As long as the logic can handle contradiction and does not explode,
I do not care how the implication is defined.

For a possible definition, see Avron's articles, e.g. "Value in four
values" and others. He defines 'a implies b' to be 1 for 0 and _|_
and b otherwise.

Is this what you mean?

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

So 1=>1 were _|_. That's too strange.

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.

Further

((A V B) a> C) a> ((A a>C) V (B a> C))

also does not (in A,B,C=T).

Maybe /\ was also prepared? Oller for example uses consensus instead of V
and gullibility instead of /\.

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

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