Re: A Simple Notation
Date: Fri, 06 Jul 2007 19:27:59 GMT
"David Cressey" <cressey73_at_verizon.net> wrote in message
> "Brian Selzer" <brian_at_selzer-software.com> wrote in message
>> "paul c" <toledobythesea_at_oohay.ac> wrote in message
>> > Brian Selzer wrote:
>> > ...
>> >> The symmetry is rather pleasing.
>> >> ...
>> > Not saying that the above comment by itself deserves to be criticized,
>> > 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
>> > as, "it gets the answer we want!").
>> As far as I can tell, David's choice of  for TRUE is arbitrary. It's
>> notation, and therefore it's his perogative to do as he pleases. But
>> is contained within the brackets is a conjunction of an arbitrary number
>> 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
>> 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