# Re: NULLs: theoretical problems?

From: Jan Hidders <hidders_at_gmail.com>
Date: Sat, 11 Aug 2007 14:14:58 -0000

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
Received on Sat Aug 11 2007 - 16:14:58 CEST

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