Re: looking for a predicate hierarchy

"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, "intuitionistic"
is a misnomer because it has been shown that, although based on
different insights, IFL is mathematically equivalent to interval-valued
fuzzy logic. IVL laso has a host of philosophical issues like how does
one substantiate *two* fuzzy membership functions ?

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.

Right, it's well known that probablity theory 'connectives' are not
truth-functional.


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.

So how does one arrive at the estimations ? Now, you have to
substantiate *two* fuzzy interval boundaries intead of one fuzzy number.
It's hardly better.


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") ! Or did I misunderstand you ?

Further

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

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

See above. That's the whole idea behind the desire to be able to handle
contradictory information: to have more than one 'designated' truth
value. the simple trick also works in the case of three-valued logic,
where depending on interpretation of the third symbol (whether it belongs
to the designated set or not), a logic can be considered paraconsistent
or just multi-valued.


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

No, it was not.



.

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