# Re: NULLs: theoretical problems?

From: V.J. Kumar <vjkmail_at_gmail.com>
Date: Sat, 11 Aug 2007 20:07:24 +0200 (CEST)

Aloha Kakuikanu <aloha.kakuikanu_at_yahoo.com> wrote in news:1186797598.242847.86560_at_q4g2000prc.googlegroups.com:

> 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:
>>
```

...
```>>
>> 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
>

Firstly, strictly speaking, a multivalued logic is not of course boolean. What you've defined is similar to Belnap's paraconsistent logic where two additional truth values are 'both' and 'undefined'. However, above, you've collapsed 'b' and 'u' to 'u' thus arriving to a three valued logic.

> 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?

You are missing the point that some multivalued logic may have multiple interpretions. I am not sure what you mean by 'consistent 4 valued logic'. A 3VL can admit an 'unknown' truth value interprtation as well as a paraconsistent one depending on what 'designated' truth values are.

>
>
Received on Sat Aug 11 2007 - 20:07:24 CEST

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