# Re: A Simple Notation

Date: Fri, 06 Jul 2007 19:27:59 GMT

Message-ID: <3Jwji.45409$5j1.33764_at_newssvr21.news.prodigy.net>

"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