RE: st: My last word on strange world

[David Kantor <kantor.d@att.net>]

At 12:37 PM 1/10/2008, Nick wrote:
> Chasing up that reference, it seems to me that in what you
> proposed, -if2- is the lesser part of the proposal, as logical
> expressions
> often arise separately from -if-.
>
> The larger part of the proposal is a three-way logic:
>
> true AND true = true
> true OR true = true
> true AND false = false
> true OR false = true
> false AND false = false
> false OR false = false
> true AND missing = missing
> true OR missing = true
> missing AND missing = missing
> missing OR missing = missing
> false AND missing = false
> false OR missing = missing
>
> In essence, there are 6 pairs here that we know already,
>
> true AND true = true
> true OR true = true
> true AND false = false
> true OR false = true
> false AND false = false
> false OR false = false
>
> and (to my first reading) two that seem "obvious"
>
> missing AND missing = missing
> missing OR missing = missing
>
> and four that don't seem totally "obvious" to me,
> at least not as a set
>
> true AND missing = missing
> true OR missing = true
> false AND missing = false
> false OR missing = missing
>
> No doubt there's a larger story here.
> [...]
Yes.
Seven years ago I did a presentation on 3-valued logic, in which I 
showed the above scheme (what you might call the standard scheme). I 
also gave another interpretation called "liberal", activated by the 
-liberal- option in the various programs of the trinary package.  The 
liberal interpretation is...
true AND missing = true
false OR missing = false
with all the others the same as the standard.

Since then, I have come to realize that these alternative options 
indicate that the results of such operations depend on what you mean 
by missing. In the standard scheme, missing signifies "unknown". The 
liberal scheme says that missing means "vacuous" -- there is nothing there.

There in another option as well, which I leave aside.

It is possible to construct logic operations (AND, OR) that include 
all the interpretations together. (You get 5-valued logic, one of my 
back-burner projects.)
Furthermore, these notions can carry over to numeric computations.
 2 + unknown = unknown
 2 + vacuous = 2
And comparisons.
 2 < unknown = unknown
 2 < vacuous  is an absurdity

(And other complications arise with comparisons.)

Computing a value is one thing, but this still leaves what to do when 
a action is conditional on a value:
list x if .

There is yet a larger story.

--David

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