Re: looking for a predicate hierarchy

"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").

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). 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 ?
;)



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 ...


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.


The definition avoids trivialization and preserves modus
ponens although it has some other drawbacks such as the deduction
system being too weak.

And modus ponens in this form

(A and (A a> B)) a> B

does not hold in 1 a> 1:

It does, see the truth table above. The definition also prevents
trivialization because the consequence relation (A a> A), ^(A a> A) |= B
does not hold for any arbitrary B any more: assign A = T, B=0 or B= _|_,
the left side has a model and the right side does not.

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.
.

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