Re: TRUE and FALSE values in the relational lattice

On Jun 19, 2:06 pm, Vadim Tropashko <vadimtro_inva..._at_yahoo.com> wrote:
> On Jun 19, 1:32 pm, Marshall <marshall.spi..._at_gmail.com> wrote:
>
> > Not to be contrary, but I've revised my thinking in that area.
> > I now equate 10 as false rather than 00. Any other relation
> > equates to true, especially, of course, 01.
>
> Propositional logic is a predicate calculus of zilliary predicates
> (zilliary predicates are the ones that have less arguments than unary
> ones:-) Therefore, associating any other relation than 01 and 00 with
> TRUE and FALSE, correspondingly, doesn't seem right.

I agree with the feeling; it is why I was saying 00 and 01 previously....
Well, I just wrote up an explanation and it wasn't at all convincing. I'll have to review my notes. Perhaps I have been overly influenced by reading lattice logic papers that don't have 00.

> > The relevance that 00 plays is in relation to existential
> > quantification.
> > I interpret existential quantification as meaning a relation is not
> > empty.
>
> Could you please be more specific?
>
> Assuming the domain x = {a,b,c,d,...} the relation "exists(x) R(x,y)"
> evaluates to
>
> R(a,y) \/ R(b,y) \/ R(c,y) \/ R(d,y) \/ ...
>
> I don't see any emptiness here.

It seems you are defining existential quantification as something that is relation valued, but I am used to thinking of it as truthvalued.

Assuming

  R(a1,b1)

Is your definition any different than

  R /\ `x=a1`

Here a1 is an attribute name and x is free, aka a parameter of the expression.

> > We can define exists(X) as:
>
> > X | 00 = 01
>
> You lost me here as well. Perhaps, you mean that the relational
> equality is an operator that evaluates to a certain relation? Can you
> be more specific?

I mean that = could be a binary
relational operator like & and | are. It evaluates to one of only two possible relation values: 10 or 01. (Presumably if I change my mind back again it would be "00 or 01.")

Thus:

(00 = 00) evaluates to 01.
(10 = 01) evaluates to 10.
(A = A) evaluates to 01.

etc.

Marshall Received on Tue Jun 19 2007 - 23:51:55 CEST