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# x*x+1=0

From: David Cressey <david_at_dcressey.com>
Date: Sun, 21 Jan 2001 13:18:47 GMT

There's an interesting book, "Laws of Form" by G. Spencer Brown, in which he explores the
consequences of a certain 4 valued Boolean logic.

x*x + 1 = 0 can be rendered in "self referential" form as x = -1/x

The assertion, "This assertion is false" has no solution, in the domain (false, true).

But if we add a "Boolean Imaginary" as the solution to the above assertion, and its complement, we get a
very "interesting" logical system, in which paradoxes are simply ordinary assertions, with imaginary
components in the solution.

I don't have the sophistication to see all the consequences, but it looked productive to me.

```--
Regards,
David Cressey
www.dcressey.com
"Jan Hidders" <hidders_at_REMOVE.THIS.win.tue.nl> wrote in message
news:94amg7\$71k\$1_at_news.tue.nl...

> wrote:
> > In article <949sjq\$k1s\$3_at_news.tue.nl>,
> >   hidders_at_win.tue.nl (Jan Hidders) wrote:
> > >  wrote:
> > > > What the closest analog of
> > > > x^2-1=0
> > > > in relational algebra would be? Does
> > > > A MULTIPLY B MULTIPLY C
> > > > have any resemblance to power 2?
> > >
> > > Not really, it is more just like multiplication. A better kind of
> > > candidate would be the powerset operation that you sometimes find in
> > > some nested relational algebras.
> >
> > Thank you for the reference. Before doing my howework, though, a newbie
> > question: Overall, are nested algebras a kind of generalization that
> > pays of?
>

> Yes, but mainly in the sense that it allows you to manipulate nested
> relations which can be a better description of your data. So the
> interest is mainly a practical one.
>

> > Is it more elegant? Does it have less number of atomic
> > operators?
>

> No, it has more; at least nesting and unnesting. Sometimes also
> other operations such as the powerset or some kind of
> recursion/iteration mechanism.
>

> > I, personally, have difficulties understanding simple
> > relational model, could I hope to get some new insights from nested one?
>

> No, things only get more complicated. But if I may ask, what is it that
> you find difficult about the flat relational model?
>

> > [...] My question is about our abilities
> > solving equations in Relational Algebra, whatever operation definitions
> > are. (Although, Distributive, Commutative, and Associative laws would
> > help:-). For example,
> >
> > x MULTIPLY A UNION B = DUM
> >
> > where A and B some table constants and x is rel var.
>

> Aha, now I am beginning to see what you mean. But I am not sure what
> your question exactly is. If you look at the equation you gave
>

>   x^2 - 1 = 0
>

> then there is no problem. It is just an equation with two solutions.
> That can also happen in the relational algebra. But I suspect what you
> wanted to talk about was the equation
>

>   x^2 + 1 = 0
>

> which under the usual interpretation does not have a solution, but
> leads to interesting theories if you assume that it does. Let's see
> what happens. First, we try to translate this into rel. algebra:
>

>   (X TIMES ({<1>} UNION {<2>})) UNION ({<3>} TIMES {<3>}) = {}
>

> Notation:
>   TIMES = the cartesian product (similar to multiplication)
>   UNION = the union (similar to addition)
>   {<x>} = singleton set with a unary tuple containing the number x
>   {} = the empty relation
>

> Assuming that this equation is solveable leads to the peculiar property
> that there will be sets that you can add a non-empty set to such that
> the result will be an empty. You might call them "negative sets" if you
> will. I doubt it they will turn out to be very useful concepts. :-)
>

> --
>   Jan Hidders
>