x*x+1=0

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


>


> PS. Could you tell your newsreader/poster your real name please? The


>     quotation of your lines now looks a bit funny. :-)