Re: looking for a predicate hierarchy

"Dmitry A. Kazakov" <mailbox@xxxxxxxxxxxxxxxxx> wrote in
news:jctw61m4fitf.or6bzp94wggz.dlg@xxxxxxxxxx:

On Fri, 22 Dec 2006 12:40:46 +0100 (CET), V.J. Kumar wrote:

"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).

not(A=>A) is false for any A. So what?

Formulas with your implication potentially cannot handle contradiction,
that's what. The whole point of having 'T' as a designated truth value
is to allow models for expressions like (F /\ ^F). Now, with your
implication it's no longer possible in the general case. In other words,
it's easy to see that <FOUR, \/, /\, ^> has a model for every formula,
whilst your <FOUR, \/, /\, ~, =>) clearly does not. It is not
surprizing: the nasty property of admitting empty models is inherited
from the '~' connective that you liked so much !



.

Relevant Pages

  • Re: Equivalence
    ... for the cases of A -> B when A is a contradiction or B is a validity. ... IMO it's an extremely useful distinction; your 'trivial' implication is ... sentences as antecedent and consequent, e.g., "If pigs have wings, ... least one interpretation or state of affairs, ...
    (sci.logic)
  • Re: Is there an error in Van Fraassens paper?
    ... and that they necessitate a contradiction for any truth-assignment to Y.. ... Let A be a contradiction and B be a contradiction. ... necessitation is NOT equivalent to implication. ...
    (sci.logic)