Re: A Simple Notation

"David Cressey" <cressey73_at_verizon.net> wrote in message news:eirji.2$475.1_at_trndny04...
>
> "Brian Selzer" <brian_at_selzer-software.com> wrote in message
> news:UHlji.18386$2v1.9600_at_newssvr14.news.prodigy.net...
>>
>> "paul c" <toledobythesea_at_oohay.ac> wrote in message
>> news:4Ifji.90354$xq1.46042_at_pd7urf1no...
>> > Brian Selzer wrote:
>> > ...
>> >> The symmetry is rather pleasing.
>> >> ...
>> >
>> > Not saying that the above comment by itself deserves to be criticized,
> but
>> > I would say that apparent lack of symmetry doesn't necessarily mean a
>> > method doesn't have same, just that we are unable to see it in some
>> > mechanical interpretation that we happen to prefer for other reasons
> (such
>> > as, "it gets the answer we want!").
>> >
>>
>> As far as I can tell, David's choice of [] for TRUE is arbitrary. It's
> his
>> notation, and therefore it's his perogative to do as he pleases. But
>> what
>> is contained within the brackets is a conjunction of an arbitrary number
> of
>> boolean values, so it makes sense to view [] as the negation of a nullary
>> product just as it makes sense to view [A] as the negation of a unary
>> product, or [A B] as the negation of a binary product, and so on. Now
>> had
>> David begun with OR and <OR>, then it would have made sense to view [] as
>> the negation of a nullary sum. A nullary sum takes on the value of the
>> additive identity which is 0 or FALSE, whereas a nullary product takes on
>> the value of the multiplicative identity which is 1 or TRUE. So,
>>
>> for OR and <OR>, [] should yield TRUE, but
>> for AND and <AND>, [] should yield FALSE
>>
>> > p
>>
>>
>
> In reaction to Brian's responses, I'm going to reformulate the notation,
> using OR and <OR> instead of AND and <AND>
>
> Thus the starting place is:
>
>
> [A B] means <NOT> (A <OR> B) in RA.
>
> Extending to 3 or more operands.....
>
> [A B C] means <NOT> (A <OR> B <OR> C) and so on.
>
> This is a classic "inverter" which I think is the same as a NAND gate.
>

Not that it matters much, but the above is similar to a NOR gate.

> 1 operand:
>
> [A] means <NOT> A as before.
>
> No operands:
>
> [] means TRUE
> [[]] means FALSE as before.
>
>
> One more item:
>
> [[]] =
>
>
> Yes, that's right, there's nothing to the right of the equal sign. At
> least at this level there is no need to introduce a third logical value to
> deal with missing items.
>
> I still haven't figured out how to make use of Bob's response regarding
> MINUS as distinct from <NOT>
>
> I guess I would want
>
> [A] to mean X MINUS A for some X that I can't figure out. Still mulling
> on this.
>
X represents the set of n-tuples of objects exemplified by the predicate of A. For each n-tuple, the proposition represented under an interpretation is assigned either true or false--that is,

for each t in X, either P(t) or ~P(t) , but not both.

The extension of the predicate of A contains only those tuples for which P(t) is true. The extension of X MINUS A contains only those tuples for which P(t) is false. Of course if the domains referenced in A are finite, then the set of n-tuples of objects exemplified by the predicate of A is also finite, so the complement of A is identical to the extension of X MINUS A. Received on Fri Jul 06 2007 - 21:27:59 CEST