# Re: NULLs: theoretical problems?

Date: Sat, 11 Aug 2007 14:14:58 -0000

Message-ID: <1186841698.402565.180140_at_d55g2000hsg.googlegroups.com>

On Aug 11, 3:59 am, Aloha Kakuikanu <aloha.kakuik..._at_yahoo.com> wrote:

> On Aug 10, 4:52 pm, "V.J. Kumar" <vjkm..._at_gmail.com> wrote:

*>
**>
**>
**> > "David Portas" <REMOVE_BEFORE_REPLYING_dpor..._at_acm.org> wrote innews:NM-dncYFOuhqBybb4p2dnAA_at_giganews.com:
**>
**> > > "paul c" <toledobythe..._at_oohay.ac> wrote in message
**> > >news:JXLui.45171$rX4.26997_at_pd7urf2no...
**>
**> > >> (even though I'm not sure in "s{X} = t{X} implies s{Y} = t{Y}"
**> > >> whether "implies" stands for logical implication.)
**>
**> > > Good catch. It seems that logical implication is not well defined for
**> > > three-value logic.
**>
**> > It is not that three-valued implication is not 'well defined' whatever it
**> > means. As a matter of fact, there are a few competing definitions to
**> > choose from, Lukaciewicz's, Kleene's and someone else's whose name I do
**> > not recall. They define implication in the usual way, with the truth
**> > table.
**>
**> I wonder if 3-rd value logic interpretation is trivial. Take any
**> boolean algebra that is more than 2 valued, and partition its elements
**> into 3 equivalence classes. For example, one may define True as
**> maximal element, False as a minimal one, and combine all the rest into
**> Unknown. For four element BA we have:
**>
**> 00 -- False
**> 01 -- Unknown
**> 10 -- Unknown
**> 11 -- True
**>
**> Sure in this model formal implication "Unknown -> Unknown" evaluates
**> to True or Unknown:
**>
**> "01 -> 01" = "01 \/ ~01" = "01 \/ 10" = "11" -- true
**>
**> on the other hand
**>
**> "01 -> 10" = "01 \/ ~10" = "01 \/ 01" = "01" -- unknown
**>
**> So the problem is to make the partition of BA elements to respect BA
**> operations, so that the later can be defined consistently. Apparently,
**> one can have consistent 4 valued logic, but not 3 valued one. Am I
**> missing anything?
*

Yes. The problem is not that there is no consistent interpretation but that there is more than one. Apart form the fact that with a truth table approach you will always have the problem that (P \/ ~P) will never evalutate to TRUE if P is not TRUE or FALSE. But that is of course the usual trade off between "what we really want" and "what can be efficiently computed".

- Jan Hidders