Re: looking for a predicate hierarchy

"Dmitry A. Kazakov" <mailbox@xxxxxxxxxxxxxxxxx> wrote in
news:1lepbpsij9lm3$.13c2j961bhgkr.dlg@xxxxxxxxxx:

On Wed, 20 Dec 2006 20:44:22 +0100 (CET), V.J. Kumar wrote:

"Dmitry A. Kazakov" <mailbox@xxxxxxxxxxxxxxxxx> wrote in
news:1iraa1mnvtcji.oh5bsnrjjcdw$.dlg@xxxxxxxxxx:

On Tue, 19 Dec 2006 21:20:53 +0100 (CET), V.J. Kumar wrote:

"Dmitry A. Kazakov" <mailbox@xxxxxxxxxxxxxxxxx> wrote in
news:12cnousl5msxh.1anmyqm356hwb$.dlg@xxxxxxxxxx:

(in logic "uncertain" is usually denoted as _|_, flipped T)

In what logic ?

Ah, there are so many. Even for a tri-state logic one could take
"contradictory" T instead of "uncertain" _|_ as the third element.

Without implication, your three-valued logic is not fully
specified.

Right. That depends on the definition of implication. (not x) V y is
well defined in tri-state logic because not _|_ = _|_. But it would
be a bad implication to take.

It the Kleene logic implication. Whether it's "good" or "bad "
surely depends on the application of such logic.

For Booleans not and ~ are equivalent.

A better one is ~xVy, where ~_|_=T. That is
not closed in tri-state logic. It is in four-state Belnap logic:

x y x=>y
------------------
0 0 1
0 1 1
0 _|_ 1
1 0 0
1 1 1
1 _|_ _|_
_|_ 0 T
_|_ 1 1
_|__|_ 1


What is the truth table for ~ ?

x ~x
---------
0 1
1 0
_|_ T
T _|_


If this is the case, your logic is trivializable becaus it has formulas
that do not have a model, and in the 4-valued logic all the formulas
are supposed to have models. Consider for example ^(A=>A) for any
valuation where ^ is the ordinary negation (0->1, 1->0, _|_->_|_, T->T).

Does the deduction theorem hold with the implication ?

That depends on the inference rules.

But talking informally, yes, that is one of the reasons why this
implication is "better." It has the properties:

_|_=>_|_ T=>T

The operation x->y defined as (not x)Vy would give only:

_|_ = (_|_->_|_) T = (T->T)

i.e. one could not reason about uncertainty and contradiction using
(not x)Vy.

Neither can one with your definition.



.

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